From stdpp Require Import prelude.
From Coq Require Import FinFun Lia Program.
From VLSM Require Import Lib.Preamble Lib.ListExtras Lib.FinExtras Lib.FinFunExtras.
From VLSM Require Import Core.VLSM Core.VLSMProjections Core.Composition Core.ProjectionTraces Core.SubProjectionTraces.
From VLSM Require Import Core.Equivocation Core.Equivocation.NoEquivocation.
From VLSM Require Import Core.Equivocators.Common Core.Equivocators.Projections Core.Equivocators.EquivocatorReplay.
From VLSM Require Import Core.Equivocators.MessageProperties Core.Equivocators.Composition.Common.
From VLSM Require Import Core.Equivocators.Composition.Projections Core.Plans.

VLSM Equivocator Full Replay Traces

In this section we show that given a trace of equivocators, one can "replay" that at the end of an existing trace, by first equivocating for each initial state and then performing each transition, but appropriately "shifted".

To make the results more general, we take the trace to be replayed to be produced by a restricted set of equivocators pre-loaded with messages satisfying some conditions.

Section all_equivocating.

Context {message : Type}
  {index : Type}
  {IndEqDec : EqDecision index}
  (IM : index -> VLSM message)
  (Hbs : forall i : index, HasBeenSentCapability (IM i))
  (Hbr : forall i : index, HasBeenReceivedCapability (IM i))
  (index_listing : list index)
  (finite_index : Listing index_listing)
  (seed : message -> Prop)
  (equivocating : list index)
  (* abbreviations *)
  (Hbo : forall i : index, HasBeenObservedCapability (IM i)
    := fun i => HasBeenObservedCapability_from_sent_received (IM i))
  (equiv_index : Type := sub_index equivocating)
  (equivocator_descriptors := equivocator_descriptors IM)
  (equivocators_state_project := equivocators_state_project IM)
  (equivocator_IM := equivocator_IM IM)
  (sub_equivocator_IM := sub_IM equivocator_IM equivocating)
  (sub_IM := sub_IM IM equivocating)
  (equivocator_descriptors_update := equivocator_descriptors_update IM)
  (proper_equivocator_descriptors := proper_equivocator_descriptors IM)
  (equivocators_trace_project := equivocators_trace_project IM)
  (Free := free_composite_vlsm IM)
  (FreeE := free_composite_vlsm equivocator_IM)
  (PreFreeE := pre_loaded_with_all_messages_vlsm FreeE)
  (FreeSubE := free_composite_vlsm sub_equivocator_IM)
  (PreFreeSubE := pre_loaded_with_all_messages_vlsm FreeSubE)
  (Hbs_sub : forall sub_i, HasBeenSentCapability (sub_IM sub_i) := sub_has_been_sent_capabilities IM equivocating Hbs)
  (SeededXE : VLSM message := seeded_equivocators_no_equivocation_vlsm IM Hbs equivocating seed)
  (equivocators_no_equivocations_vlsm := equivocators_no_equivocations_vlsm IM Hbs)
.

Lemma SeededXE_Free_full_projection
  (Hseed : forall m, seed m -> valid_message_prop FreeE m)
  : VLSM_full_projection SeededXE
    (composite_vlsm equivocator_IM (free_constraint equivocator_IM))
    (lift_sub_label equivocator_IM equivocating) (lift_sub_state equivocator_IM equivocating).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
VLSM_full_projection SeededXE
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_label equivocator_IM equivocating)
  (lift_sub_state equivocator_IM equivocating)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
VLSM_full_projection SeededXE
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_label equivocator_IM equivocating)
  (lift_sub_state equivocator_IM equivocating)
  apply basic_VLSM_full_projection; intro; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
HomY: option_valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) om
vvalid
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_label equivocator_IM equivocating l)
  (lift_sub_state equivocator_IM equivocating s, om)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition SeededXE l 
  (s, om) 
  (s', om')
vtransition
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_label equivocator_IM equivocating l)
  (lift_sub_state equivocator_IM equivocating s, om) =
(lift_sub_state equivocator_IM equivocating s', om')
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
s: state
H: vinitial_state_prop SeededXE s
vinitial_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
HmX: vinitial_message_prop SeededXE m
valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) m
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
HomY: option_valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) om
vvalid
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_label equivocator_IM equivocating l)
  (lift_sub_state equivocator_IM equivocating s, om) split; [|exact I].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
HomY: option_valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) om
composite_valid equivocator_IM
  (lift_sub_label equivocator_IM equivocating l)
  (lift_sub_state equivocator_IM equivocating s, om)
    apply lift_sub_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
HomY: option_valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) om
composite_valid
  (SubProjectionTraces.sub_IM equivocator_IM
     equivocating) l (s, om) apply Hv.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition SeededXE l 
  (s, om) 
  (s', om')
vtransition
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_label equivocator_IM equivocating l)
  (lift_sub_state equivocator_IM equivocating s, om) =
(lift_sub_state equivocator_IM equivocating s', om') apply lift_sub_transition.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition SeededXE l 
  (s, om) 
  (s', om')
composite_transition
  (SubProjectionTraces.sub_IM equivocator_IM
     equivocating) l (s, om) = (s', om') apply H.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
s: state
H: vinitial_state_prop SeededXE s
vinitial_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s) apply (lift_sub_state_initial equivocator_IM).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
s: state
H: vinitial_state_prop SeededXE s
composite_initial_state_prop
  (SubProjectionTraces.sub_IM equivocator_IM
     equivocating) s assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
HmX: vinitial_message_prop SeededXE m
valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) m destruct HmX as [Hinit|Hseeded]; [|apply Hseed; assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
Hinit: initial_message_prop m
valid_message_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM)) m
    apply initial_message_is_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
Hinit: initial_message_prop m
initial_message_prop m
    destruct Hinit as [i Him].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
i: sub_index equivocating
Him: ∃ mi : vinitial_message
         (SubProjectionTraces.sub_IM
         (Common.equivocator_IM IM) equivocating
         i), 
  ` mi = m
initial_message_prop m
    exists (proj1_sig i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
Hseed: ∀ m : message,
  seed m → valid_message_prop FreeE m
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop
  (composite_vlsm equivocator_IM
     (free_constraint equivocator_IM))
  (lift_sub_state equivocator_IM equivocating s)
i: sub_index equivocating
Him: ∃ mi : vinitial_message
         (SubProjectionTraces.sub_IM
         (Common.equivocator_IM IM) equivocating
         i), 
  ` mi = m
∃ mi : vinitial_message (equivocator_IM (` i)),
  ` mi = m assumption.
Qed.

Given a base_state to replay on, the replay label corresponding to a given transition label is obtained as the equivocator_state_append_label.

Definition lift_equivocators_sub_label_to
  (base_s : composite_state equivocator_IM)
  (l : composite_label sub_equivocator_IM)
  : composite_label equivocator_IM
  :=
  let (sub_i, li) := l in
  let i := proj1_sig sub_i in
  existT i (equivocator_state_append_label (IM i) (base_s i) li).

Given a base_state to replay on, the replay state corresponding to a destination state in a transition by appending its components to the base state using equivocator_state_append.

Definition lift_equivocators_sub_state_to
  (base_s : composite_state equivocator_IM)
  (s : composite_state sub_equivocator_IM)
  : composite_state equivocator_IM
  := fun i =>
    match @decide  (sub_index_prop equivocating i) (sub_index_prop_dec equivocating i) with
    | left e =>  equivocator_state_append (base_s i) (s (dec_exist _ i e))
    | _ => base_s i
    end.

Lemma lift_equivocators_sub_state_to_sub
  (base_s : composite_state equivocator_IM)
  (s : composite_state sub_equivocator_IM)
  i
  (Hi : sub_index_prop equivocating i)
  : lift_equivocators_sub_state_to base_s s i =  equivocator_state_append (base_s i) (s (dec_exist _ i Hi)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
lift_equivocators_sub_state_to base_s s i =
equivocator_state_append (base_s i)
  (s (dec_exist (sub_index_prop equivocating) i Hi))
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
lift_equivocators_sub_state_to base_s s i =
equivocator_state_append (base_s i)
  (s (dec_exist (sub_index_prop equivocating) i Hi))
  unfold lift_equivocators_sub_state_to.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
match decide (sub_index_prop equivocating i) with
| left e =>
    equivocator_state_append (base_s i)
      (s (dec_exist (sub_index_prop equivocating) i e))
| in_right => base_s i
end =
equivocator_state_append (base_s i)
  (s (dec_exist (sub_index_prop equivocating) i Hi))
  destruct (decide _); [|contradiction].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
Hi, s0: sub_index_prop equivocating i
equivocator_state_append (base_s i)
  (s (dec_exist (sub_index_prop equivocating) i s0)) =
equivocator_state_append (base_s i)
  (s (dec_exist (sub_index_prop equivocating) i Hi))
  replace ((s (dec_exist (sub_index_prop equivocating) i s0))) with (s (dec_exist (sub_index_prop equivocating) i Hi))
  ; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
Hi, s0: sub_index_prop equivocating i
s (dec_exist (sub_index_prop equivocating) i Hi) =
s (dec_exist (sub_index_prop equivocating) i s0)
  apply sub_IM_state_pi.
Qed.

Lemma lift_equivocators_sub_state_to_size
  (base_s : composite_state equivocator_IM)
  (s : composite_state sub_equivocator_IM)
  : forall i,
    equivocator_state_n (base_s i) <= equivocator_state_n (lift_equivocators_sub_state_to base_s s i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
∀ i : index,
  equivocator_state_n (base_s i)
  ≤ equivocator_state_n
      (lift_equivocators_sub_state_to base_s s i)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
∀ i : index,
  equivocator_state_n (base_s i)
  ≤ equivocator_state_n
      (lift_equivocators_sub_state_to base_s s i)
  intro i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
equivocator_state_n (base_s i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to base_s s i)
  unfold lift_equivocators_sub_state_to.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
equivocator_state_n (base_s i)
≤ equivocator_state_n
    match decide (sub_index_prop equivocating i) with
    | left e =>
        equivocator_state_append (base_s i)
          (s
             (dec_exist (sub_index_prop equivocating)
                i e))
    | in_right => base_s i
    end
  destruct (decide _); [|lia].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
s0: sub_index_prop equivocating i
equivocator_state_n (base_s i)
≤ equivocator_state_n
    (equivocator_state_append (base_s i)
       (s
          (dec_exist (sub_index_prop equivocating) i
             s0)))
  rewrite equivocator_state_append_size.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
base_s: composite_state equivocator_IM
s: composite_state sub_equivocator_IM
i: index
s0: sub_index_prop equivocating i
equivocator_state_n (base_s i)
≤ equivocator_state_n (base_s i) +
  equivocator_state_n
    (s (dec_exist (sub_index_prop equivocating) i s0)) lia.
Qed.

The plan item corresponding to an initial state equivocation

Definition initial_new_machine_transition_item
  (is : composite_state sub_equivocator_IM)
  (eqv : equiv_index)
  : composite_plan_item equivocator_IM
  :=
  let i := proj1_sig eqv in
  let seqv := is eqv in
  let new_l :=
    (existT i (Spawn (equivocator_state_zero seqv)))
    in
  @Build_plan_item message (composite_type equivocator_IM) new_l None.

Command for equivocating all states of an initial composite state.

Definition spawn_initial_state
  (is : composite_state sub_equivocator_IM)
  : composite_plan equivocator_IM
  := map (initial_new_machine_transition_item is) (@sub_index_listing _ _ equivocating index_listing).

Definition replayed_initial_state_from full_replay_state is :=
  fst (composite_apply_plan equivocator_IM full_replay_state (spawn_initial_state is)).

The final state obtained after replaying an initial state is precisely the lifting of that initial state over the given base state.

Lemma replayed_initial_state_from_lift
  (full_replay_state : composite_state equivocator_IM)
  (is : composite_state sub_equivocator_IM)
  (His : composite_initial_state_prop sub_equivocator_IM is)
  : finite_trace_last full_replay_state (replayed_initial_state_from full_replay_state is)
    = lift_equivocators_sub_state_to full_replay_state is.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is
  cut (forall l (Hincl : incl l (@sub_index_listing _ _ equivocating index_listing)) (Hnodup : NoDup l),
    let tr_full_replay_is :=
      composite_apply_plan equivocator_IM full_replay_state
        (map (initial_new_machine_transition_item is)
          l) in
    (∀ i : index,
      tr_full_replay_is.2 i =
      match @decide  (sub_index_prop equivocating i) (sub_index_prop_dec equivocating i) with
      | left e =>
        let eqv := (dec_exist _ i e) in
        if (decide (eqv ∈ l)) then equivocator_state_append (full_replay_state i) (is eqv)
        else full_replay_state i
      | _ =>  full_replay_state i
      end
    )).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
(∀ l : list (sub_index equivocating),
   incl l (sub_index_listing equivocating)
   → NoDup l
     → ∀ (tr_full_replay_is :=
          composite_apply_plan equivocator_IM
            full_replay_state
            (map
               (initial_new_machine_transition_item is)
               l)) (i : index),
         tr_full_replay_is.2 i =
         match
           decide (sub_index_prop equivocating i)
         with
         | left e =>
             let eqv :=
               dec_exist (sub_index_prop equivocating)
                 i e in
             if decide (eqv ∈ l)
             then
              equivocator_state_append
                (full_replay_state i) (is eqv)
             else full_replay_state i
         | in_right => full_replay_state i
         end)
→ finite_trace_last full_replay_state
    (replayed_initial_state_from full_replay_state is) =
  lift_equivocators_sub_state_to full_replay_state is
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
∀ l : list (sub_index equivocating),
  incl l (sub_index_listing equivocating)
  → NoDup l
    → ∀ (tr_full_replay_is :=
         composite_apply_plan equivocator_IM
         full_replay_state
         (map 
         (initial_new_machine_transition_item is) l)) 
        (i : index),
        tr_full_replay_is.2 i =
        match
         decide (sub_index_prop equivocating i)
        with
        | left e =>
         let eqv :=
         dec_exist (sub_index_prop equivocating) i e
         in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
        | in_right => full_replay_state i
        end
  {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
(∀ l : list (sub_index equivocating),
   incl l (sub_index_listing equivocating)
   → NoDup l
     → ∀ (tr_full_replay_is :=
          composite_apply_plan equivocator_IM
            full_replay_state
            (map
               (initial_new_machine_transition_item is)
               l)) (i : index),
         tr_full_replay_is.2 i =
         match
           decide (sub_index_prop equivocating i)
         with
         | left e =>
             let eqv :=
               dec_exist (sub_index_prop equivocating)
                 i e in
             if decide (eqv ∈ l)
             then
              equivocator_state_append
                (full_replay_state i) (is eqv)
             else full_replay_state i
         | in_right => full_replay_state i
         end)
→ finite_trace_last full_replay_state
    (replayed_initial_state_from full_replay_state is) =
  lift_equivocators_sub_state_to full_replay_state is intros H.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ l : list (sub_index equivocating),
  incl l (sub_index_listing equivocating)
  → NoDup l
    → ∀ (tr_full_replay_is :=
         composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)) 
        (i : index),
        tr_full_replay_is.2 i =
        match
         decide (sub_index_prop equivocating i)
        with
        | left e =>
         let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
        | in_right => full_replay_state i
        end
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is specialize (H _ (incl_refl _)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: NoDup (sub_index_listing equivocating)
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         (sub_index_listing equivocating))) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if
         decide
         (eqv ∈ sub_index_listing equivocating)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is
    spec H.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: NoDup (sub_index_listing equivocating)
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         (sub_index_listing equivocating))) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if
         decide
         (eqv ∈ sub_index_listing equivocating)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
NoDup (sub_index_listing equivocating)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map 
        (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if
       decide
         (eqv ∈ sub_index_listing equivocating)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is
    {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: NoDup (sub_index_listing equivocating)
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         (sub_index_listing equivocating))) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if
         decide
         (eqv ∈ sub_index_listing equivocating)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
NoDup (sub_index_listing equivocating) apply NoDup_ListNoDup.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: NoDup (sub_index_listing equivocating)
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         (sub_index_listing equivocating))) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if
         decide
         (eqv ∈ sub_index_listing equivocating)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
List.NoDup (sub_index_listing equivocating) apply (finite_sub_index equivocating finite_index). }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map 
        (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if
       decide
         (eqv ∈ sub_index_listing equivocating)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is
    unfold replayed_initial_state_from, composite_apply_plan.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map 
        (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if
       decide
         (eqv ∈ sub_index_listing equivocating)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
finite_trace_last full_replay_state
  (_apply_plan full_replay_state
     (spawn_initial_state is)).1 =
lift_equivocators_sub_state_to full_replay_state is rewrite _apply_plan_last.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map 
        (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if
       decide
         (eqv ∈ sub_index_listing equivocating)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
(_apply_plan full_replay_state
   (spawn_initial_state is)).2 =
lift_equivocators_sub_state_to full_replay_state is
    apply functional_extensionality_dep.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map 
        (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if
       decide
         (eqv ∈ sub_index_listing equivocating)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
∀ x : index,
  (_apply_plan full_replay_state
     (spawn_initial_state is)).2 x =
  lift_equivocators_sub_state_to full_replay_state is
    x intro i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map 
        (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if
       decide
         (eqv ∈ sub_index_listing equivocating)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
i: index
(_apply_plan full_replay_state
   (spawn_initial_state is)).2 i =
lift_equivocators_sub_state_to full_replay_state is i
    spec H i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
H: (composite_apply_plan equivocator_IM
   full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e
      in
    if
     decide (eqv ∈ sub_index_listing equivocating)
    then
     equivocator_state_append
       (full_replay_state i) 
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end
(_apply_plan full_replay_state
   (spawn_initial_state is)).2 i =
lift_equivocators_sub_state_to full_replay_state is i unfold composite_apply_plan in H.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e
        ∈ 
        sub_index_listing equivocating)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist 
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
(_apply_plan full_replay_state
   (spawn_initial_state is)).2 i =
lift_equivocators_sub_state_to full_replay_state is i unfold spawn_initial_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e
        ∈ 
        sub_index_listing equivocating)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist 
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
(_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
lift_equivocators_sub_state_to full_replay_state is i
    simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e
        ∈ 
        sub_index_listing equivocating)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist 
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
(_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
lift_equivocators_sub_state_to full_replay_state is i rewrite H.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e
        ∈ 
        sub_index_listing equivocating)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist 
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ sub_index_listing equivocating)
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end =
lift_equivocators_sub_state_to full_replay_state is i
    unfold lift_equivocators_sub_state_to.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e
        ∈ 
        sub_index_listing equivocating)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist 
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ sub_index_listing equivocating)
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end =
match decide (sub_index_prop equivocating i) with
| left e =>
    equivocator_state_append (full_replay_state i)
      (is
         (dec_exist (sub_index_prop equivocating) i e))
| in_right => full_replay_state i
end
    destruct (decide _); [|reflexivity].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
s: sub_index_prop equivocating i
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ sub_index_listing equivocating)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ sub_index_listing equivocating)
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) =
equivocator_state_append (full_replay_state i)
  (is (dec_exist (sub_index_prop equivocating) i s))
    rewrite decide_True; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
s: sub_index_prop equivocating i
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ sub_index_listing equivocating)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
dec_exist (sub_index_prop equivocating) i s
∈ sub_index_listing equivocating
    apply elem_of_list_In.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
i: index
s: sub_index_prop equivocating i
H: (_apply_plan full_replay_state
   (map (initial_new_machine_transition_item is)
      (sub_index_listing equivocating))).2 i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ sub_index_listing equivocating)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
In (dec_exist (sub_index_prop equivocating) i s)
  (sub_index_listing equivocating)
    apply (finite_sub_index equivocating finite_index).
  }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
∀ l : list (sub_index equivocating),
  incl l (sub_index_listing equivocating)
  → NoDup l
    → ∀ (tr_full_replay_is :=
         composite_apply_plan equivocator_IM
           full_replay_state
           (map
              (initial_new_machine_transition_item is)
              l)) (i : index),
        tr_full_replay_is.2 i =
        match
          decide (sub_index_prop equivocating i)
        with
        | left e =>
            let eqv :=
              dec_exist (sub_index_prop equivocating)
                i e in
            if decide (eqv ∈ l)
            then
             equivocator_state_append
               (full_replay_state i) (is eqv)
            else full_replay_state i
        | in_right => full_replay_state i
        end
  induction l using rev_ind; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hincl: incl [] (sub_index_listing equivocating)
Hnodup: NoDup []
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) []): (list transition_item * state)%type
i: index
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ [])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ NoDup l
  → ∀ (tr_full_replay_is :=
       composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)) 
      (i : index),
      tr_full_replay_is.2 i =
      match
        decide (sub_index_prop equivocating i)
      with
      | left e =>
         let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
      | in_right => full_replay_state i
      end
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append 
       (full_replay_state i) 
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hincl: incl [] (sub_index_listing equivocating)
Hnodup: NoDup []
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) []): (list transition_item * state)%type
i: index
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ [])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end destruct (decide _); [|reflexivity].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hincl: incl [] (sub_index_listing equivocating)
Hnodup: NoDup []
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) []): (list transition_item * state)%type
i: index
s: sub_index_prop equivocating i
tr_full_replay_is.2 i =
(let eqv :=
   dec_exist (sub_index_prop equivocating) i s in
 if decide (eqv ∈ [])
 then
  equivocator_state_append (full_replay_state i)
    (is eqv)
 else full_replay_state i)
    rewrite decide_False; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hincl: incl [] (sub_index_listing equivocating)
Hnodup: NoDup []
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) []): (list transition_item * state)%type
i: index
s: sub_index_prop equivocating i
dec_exist (sub_index_prop equivocating) i s ∉ [] intro Hin.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hincl: incl [] (sub_index_listing equivocating)
Hnodup: NoDup []
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) []): (list transition_item * state)%type
i: index
s: sub_index_prop equivocating i
Hin: dec_exist (sub_index_prop equivocating) i s ∈ []
False inversion Hin.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ NoDup l
  → ∀ (tr_full_replay_is :=
       composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)) 
      (i : index),
      tr_full_replay_is.2 i =
      match
        decide (sub_index_prop equivocating i)
      with
      | left e =>
         let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
      | in_right => full_replay_state i
      end
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end spec IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ NoDup l
  → ∀ (tr_full_replay_is :=
       composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)) 
      (i : index),
      tr_full_replay_is.2 i =
      match
        decide (sub_index_prop equivocating i)
      with
      | left e =>
         let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
      | in_right => full_replay_state i
      end
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
incl l (sub_index_listing equivocating)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: NoDup l
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         l)) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if decide (eqv ∈ l)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append 
       (full_replay_state i) 
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ NoDup l
  → ∀ (tr_full_replay_is :=
       composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)) 
      (i : index),
      tr_full_replay_is.2 i =
      match
        decide (sub_index_prop equivocating i)
      with
      | left e =>
         let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
      | in_right => full_replay_state i
      end
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
incl l (sub_index_listing equivocating) apply incl_app_inv in Hincl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ NoDup l
  → ∀ (tr_full_replay_is :=
       composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)) 
      (i : index),
      tr_full_replay_is.2 i =
      match
        decide (sub_index_prop equivocating i)
      with
      | left e =>
         let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
         if decide (eqv ∈ l)
         then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
         else full_replay_state i
      | in_right => full_replay_state i
      end
Hincl: incl l (sub_index_listing equivocating)
∧ incl [x] (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
incl l (sub_index_listing equivocating) apply Hincl. }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: NoDup l
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         l)) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if decide (eqv ∈ l)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end
    spec IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: NoDup l
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         l)) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if decide (eqv ∈ l)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
NoDup l
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if decide (eqv ∈ l)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append 
       (full_replay_state i) 
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: NoDup l
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         l)) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if decide (eqv ∈ l)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
NoDup l  apply NoDup_app in Hnodup.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup l
∧ (∀ x0 : sub_index equivocating,
     x0 ∈ l → x0 ∉ [x]) ∧ 
  NoDup [x]
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: NoDup l
→ ∀ (tr_full_replay_is :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map
         (initial_new_machine_transition_item is)
         l)) 
    (i : index),
    tr_full_replay_is.2 i =
    match
      decide (sub_index_prop equivocating i)
    with
    | left e =>
        let eqv :=
         dec_exist 
         (sub_index_prop equivocating) i e in
        if decide (eqv ∈ l)
        then
         equivocator_state_append
         (full_replay_state i) 
         (is eqv)
        else full_replay_state i
    | in_right => full_replay_state i
    end
NoDup l apply Hnodup.  }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
tr_full_replay_is:= composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item
        is) 
     (l ++ [x])): (list transition_item * state)%type
i: index
IHl: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if decide (eqv ∈ l)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
tr_full_replay_is.2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end
    subst tr_full_replay_is.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
IHl: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if decide (eqv ∈ l)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
(composite_apply_plan equivocator_IM full_replay_state
   (map (initial_new_machine_transition_item is)
      (l ++ [x]))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end
    rewrite map_app.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
IHl: ∀ (tr_full_replay_is :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)) 
  (i : index),
  tr_full_replay_is.2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      let eqv :=
        dec_exist 
         (sub_index_prop equivocating) i e in
      if decide (eqv ∈ l)
      then
       equivocator_state_append
         (full_replay_state i) 
         (is eqv)
      else full_replay_state i
  | in_right => full_replay_state i
  end
(composite_apply_plan equivocator_IM full_replay_state
   (map (initial_new_machine_transition_item is) l ++
    map (initial_new_machine_transition_item is) [x])).2
  i =
match decide (sub_index_prop equivocating i) with
| left e =>
    let eqv :=
      dec_exist (sub_index_prop equivocating) i e in
    if decide (eqv ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is eqv)
    else full_replay_state i
| in_right => full_replay_state i
end simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
IHl: ∀ i : index,
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)).2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      if
       decide
         (dec_exist
         (sub_index_prop equivocating) i e ∈ l)
      then
       equivocator_state_append
         (full_replay_state i)
         (is
         (dec_exist
         (sub_index_prop equivocating) i e))
      else full_replay_state i
  | in_right => full_replay_state i
  end
(composite_apply_plan equivocator_IM full_replay_state
   (map (initial_new_machine_transition_item is) l ++
    [initial_new_machine_transition_item is x])).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    rewrite (composite_apply_plan_app equivocator_IM).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
IHl: ∀ i : index,
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)).2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      if
       decide
         (dec_exist
         (sub_index_prop equivocating) i e ∈ l)
      then
       equivocator_state_append
         (full_replay_state i)
         (is
         (dec_exist
         (sub_index_prop equivocating) i e))
      else full_replay_state i
  | in_right => full_replay_state i
  end
(let (aitems, afinal) :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map (initial_new_machine_transition_item is) l) in
 let (a'items, a'final) :=
   composite_apply_plan equivocator_IM afinal
     [initial_new_machine_transition_item is x] in
 (aitems ++ a'items, a'final)).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
IHl: ∀ i : index,
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)).2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      if
       decide
         (dec_exist
         (sub_index_prop equivocating) i e ∈ l)
      then
       equivocator_state_append
         (full_replay_state i)
         (is
         (dec_exist
         (sub_index_prop equivocating) i e))
      else full_replay_state i
  | in_right => full_replay_state i
  end
(let (aitems, afinal) :=
   composite_apply_plan equivocator_IM
     full_replay_state
     (map (initial_new_machine_transition_item is) l) in
 (aitems ++
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM afinal (` x)
         (equivocator_state_extend (afinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}],
 state_update equivocator_IM afinal (` x)
   (equivocator_state_extend (afinal (` x))
      (equivocator_state_zero (is x))))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    destruct (composite_apply_plan _ _ _) as (aitems, afinal).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
aitems: list transition_item
afinal: state
IHl: ∀ i : index,
  (aitems, afinal).2 i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      if
       decide
         (dec_exist
         (sub_index_prop equivocating) i e ∈ l)
      then
       equivocator_state_append
         (full_replay_state i)
         (is
         (dec_exist
         (sub_index_prop equivocating) i e))
      else full_replay_state i
  | in_right => full_replay_state i
  end
(aitems ++
 [{|
    l :=
      existT (` x)
        (Spawn (equivocator_state_zero (is x)));
    input := None;
    destination :=
      state_update equivocator_IM afinal (` x)
        (equivocator_state_extend (afinal (` x))
           (equivocator_state_zero (is x)));
    output := None
  |}],
state_update equivocator_IM afinal (` x)
  (equivocator_state_extend (afinal (` x))
     (equivocator_state_zero (is x)))).2 i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
IHl: ∀ i : index,
  afinal i =
  match
    decide (sub_index_prop equivocating i)
  with
  | left e =>
      if
       decide
         (dec_exist
         (sub_index_prop equivocating) i e ∈ l)
      then
       equivocator_state_append
         (full_replay_state i)
         (is
         (dec_exist
         (sub_index_prop equivocating) i e))
      else full_replay_state i
  | in_right => full_replay_state i
  end
state_update equivocator_IM afinal (` x)
  (equivocator_state_extend (afinal (` x))
     (equivocator_state_zero (is x))) i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    spec IHl i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
IHl: afinal i =
match
  decide (sub_index_prop equivocating i)
with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e ∈ l)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
state_update equivocator_IM afinal (` x)
  (equivocator_state_extend (afinal (` x))
     (equivocator_state_zero (is x))) i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end destruct_dec_sig x ix Hix Heqx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
Hincl: incl (l ++ [x])
  (sub_index_listing equivocating)
Hnodup: NoDup (l ++ [x])
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
IHl: afinal i =
match
  decide (sub_index_prop equivocating i)
with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e ∈ l)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
ix: index
Hix: sub_index_prop equivocating ix
Heqx: x =
dec_exist (sub_index_prop equivocating) ix Hix
state_update equivocator_IM afinal (` x)
  (equivocator_state_extend (afinal (` x))
     (equivocator_state_zero (is x))) i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++ [x])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end subst x.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
IHl: afinal i =
match
  decide (sub_index_prop equivocating i)
with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e ∈ l)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
state_update equivocator_IM afinal
  (` (dec_exist (sub_index_prop equivocating) ix Hix))
  (equivocator_state_extend
     (afinal
        (`
         (dec_exist (sub_index_prop equivocating) ix
            Hix)))
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++
          [dec_exist (sub_index_prop equivocating) ix
             Hix])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
IHl: afinal i =
match
  decide (sub_index_prop equivocating i)
with
| left e =>
    if
     decide
       (dec_exist 
         (sub_index_prop equivocating) i e ∈ l)
    then
     equivocator_state_append
       (full_replay_state i)
       (is
         (dec_exist
         (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
match decide (sub_index_prop equivocating i) with
| left e =>
    if
     decide
       (dec_exist (sub_index_prop equivocating) i e
        ∈ l ++
          [dec_exist (sub_index_prop equivocating) ix
             Hix])
    then
     equivocator_state_append (full_replay_state i)
       (is
          (dec_exist (sub_index_prop equivocating) i e))
    else full_replay_state i
| in_right => full_replay_state i
end
    destruct (decide _).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
n: ¬ sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
state_update equivocator_IM afinal ix
  (equivocator_state_extend 
     (afinal ix)
     (equivocator_state_zero
        (is
         (dec_exist 
         (sub_index_prop equivocating) ix Hix)))) i =
full_replay_state i
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) destruct (decide (ix = i)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
e: ix = i
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
n: ix ≠ i
state_update equivocator_IM afinal ix
  (equivocator_state_extend 
     (afinal ix)
     (equivocator_state_zero
        (is
         (dec_exist 
         (sub_index_prop equivocating) ix Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append 
    (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
      *message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
e: ix = i
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) subst ix.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
state_update equivocator_IM afinal i
  (equivocator_state_extend (afinal i)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) i
              Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) i Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) rewrite state_update_eq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
equivocator_state_extend (afinal i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) i Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
        rewrite decide_False in IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
equivocator_state_extend (afinal i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) i Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
dec_exist (sub_index_prop equivocating) i s ∉ l
        2: {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
dec_exist (sub_index_prop equivocating) i s ∉ l intro Heqv.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
Heqv: dec_exist (sub_index_prop equivocating) i s ∈ l
False
          apply NoDup_app, proj2, proj1 in Hnodup.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: ∀ x : sub_index equivocating,
  x ∈ l
  → x
    ∉ [dec_exist
         (sub_index_prop equivocating) i Hix]
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
Heqv: dec_exist (sub_index_prop equivocating) i s ∈ l
False
          elim (Hnodup _ Heqv).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: ∀ x : sub_index equivocating,
  x ∈ l
  → x
    ∉ [dec_exist
         (sub_index_prop equivocating) i Hix]
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
Heqv: dec_exist (sub_index_prop equivocating) i s ∈ l
dec_exist (sub_index_prop equivocating) i s
∈ [dec_exist (sub_index_prop equivocating) i Hix]
          replace (dec_exist _ i s) with (dec_exist _ i Hix (P_dec := sub_index_prop_dec equivocating))
            by (apply dec_sig_eq_iff; reflexivity).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: ∀ x : sub_index equivocating,
  x ∈ l
  → x
    ∉ [dec_exist
         (sub_index_prop equivocating) i Hix]
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
Heqv: dec_exist (sub_index_prop equivocating) i s ∈ l
dec_exist (sub_index_prop equivocating) i Hix
∈ [dec_exist (sub_index_prop equivocating) i Hix]
          left.
        }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
equivocator_state_extend (afinal i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) i Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
        rewrite IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
equivocator_state_extend (full_replay_state i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) i Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
        rewrite decide_True.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
equivocator_state_extend (full_replay_state i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
equivocator_state_append (full_replay_state i)
  (is (dec_exist (sub_index_prop equivocating) i s))
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
dec_exist (sub_index_prop equivocating) i s
∈ l ++ [dec_exist (sub_index_prop equivocating) i Hix]
        --message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
equivocator_state_extend (full_replay_state i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
equivocator_state_append (full_replay_state i)
  (is (dec_exist (sub_index_prop equivocating) i s)) replace (is (dec_exist _ i s)) with (is (dec_exist _ i Hix (P_dec := sub_index_prop_dec equivocating)))
          by apply sub_IM_state_pi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
equivocator_state_extend (full_replay_state i)
  (equivocator_state_zero
     (is
        (dec_exist (sub_index_prop equivocating) i Hix))) =
equivocator_state_append (full_replay_state i)
  (is (dec_exist (sub_index_prop equivocating) i Hix))
          rewrite equivocator_state_append_singleton_is_extend; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
is_singleton_state (IM i)
  (is (dec_exist (sub_index_prop equivocating) i Hix))
          apply (His (dec_exist (sub_index_prop equivocating) i Hix)).
        --message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
dec_exist (sub_index_prop equivocating) i s
∈ l ++ [dec_exist (sub_index_prop equivocating) i Hix] apply elem_of_app.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
dec_exist (sub_index_prop equivocating) i s ∈ l
∨ dec_exist (sub_index_prop equivocating) i s
  ∈ [dec_exist (sub_index_prop equivocating) i Hix] right.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
dec_exist (sub_index_prop equivocating) i s
∈ [dec_exist (sub_index_prop equivocating) i Hix]
          replace (dec_exist _ i s) with (dec_exist _ i Hix (P_dec := sub_index_prop_dec equivocating))
            by (apply dec_sig_eq_iff; reflexivity).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
i: index
Hix: sub_index_prop equivocating i
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
  (sub_index_listing equivocating)
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) i Hix])
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
dec_exist (sub_index_prop equivocating) i Hix
∈ [dec_exist (sub_index_prop equivocating) i Hix]
          left.
      *message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
n: ix ≠ i
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) rewrite state_update_neq by congruence.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
IHl: afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l)
 then
  equivocator_state_append 
    (full_replay_state i)
    (is
       (dec_exist 
         (sub_index_prop equivocating) i s))
 else full_replay_state i)
n: ix ≠ i
afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
        destruct (decide (_ ∈ l)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
e: dec_exist (sub_index_prop equivocating) i s ∈ l
IHl: afinal i =
equivocator_state_append 
  (full_replay_state i)
  (is
     (dec_exist 
        (sub_index_prop equivocating) i s))
n: ix ≠ i
afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append 
    (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i)
        --message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
e: dec_exist (sub_index_prop equivocating) i s ∈ l
IHl: afinal i =
equivocator_state_append 
  (full_replay_state i)
  (is
     (dec_exist 
        (sub_index_prop equivocating) i s))
n: ix ≠ i
afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) rewrite decide_True; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
e: dec_exist (sub_index_prop equivocating) i s ∈ l
IHl: afinal i =
equivocator_state_append 
  (full_replay_state i)
  (is
     (dec_exist 
        (sub_index_prop equivocating) i s))
n: ix ≠ i
dec_exist (sub_index_prop equivocating) i s
∈ l ++
  [dec_exist (sub_index_prop equivocating) ix Hix]
          apply elem_of_app.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
e: dec_exist (sub_index_prop equivocating) i s ∈ l
IHl: afinal i =
equivocator_state_append 
  (full_replay_state i)
  (is
     (dec_exist 
        (sub_index_prop equivocating) i s))
n: ix ≠ i
dec_exist (sub_index_prop equivocating) i s ∈ l
∨ dec_exist (sub_index_prop equivocating) i s
  ∈ [dec_exist (sub_index_prop equivocating) ix Hix] left.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
e: dec_exist (sub_index_prop equivocating) i s ∈ l
IHl: afinal i =
equivocator_state_append 
  (full_replay_state i)
  (is
     (dec_exist 
        (sub_index_prop equivocating) i s))
n: ix ≠ i
dec_exist (sub_index_prop equivocating) i s ∈ l assumption.
        --message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
afinal i =
(if
  decide
    (dec_exist (sub_index_prop equivocating) i s
     ∈ l ++
       [dec_exist (sub_index_prop equivocating) ix Hix])
 then
  equivocator_state_append (full_replay_state i)
    (is (dec_exist (sub_index_prop equivocating) i s))
 else full_replay_state i) rewrite decide_False; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
dec_exist (sub_index_prop equivocating) i s
∉ l ++
  [dec_exist (sub_index_prop equivocating) ix Hix]
          intro Hin.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
Hin: dec_exist (sub_index_prop equivocating) i s
∈ l ++
  [dec_exist 
     (sub_index_prop equivocating) ix Hix]
False apply elem_of_app in Hin.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
Hin: dec_exist (sub_index_prop equivocating) i s ∈ l
∨ dec_exist (sub_index_prop equivocating) i s
  ∈ [dec_exist 
       (sub_index_prop equivocating) ix Hix]
False
          destruct Hin as [Hin | Hx]; [contradiction|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
Hx: dec_exist (sub_index_prop equivocating) i s
∈ [dec_exist (sub_index_prop equivocating) ix Hix]
False
          apply elem_of_list_singleton in Hx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
Hx: dec_exist (sub_index_prop equivocating) i s =
dec_exist (sub_index_prop equivocating) ix Hix
False
          apply dsig_eq in Hx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
Hx: ` (dec_exist (sub_index_prop equivocating) i s) =
`
(dec_exist (sub_index_prop equivocating) ix Hix)
False simpl in Hx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
s: sub_index_prop equivocating i
n0: dec_exist (sub_index_prop equivocating) i s ∉ l
IHl: afinal i = full_replay_state i
n: ix ≠ i
Hx: i = ix
False congruence.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
n: ¬ sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
state_update equivocator_IM afinal ix
  (equivocator_state_extend (afinal ix)
     (equivocator_state_zero
        (is
           (dec_exist (sub_index_prop equivocating) ix
              Hix)))) i = full_replay_state i rewrite state_update_neq; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
ix: index
Hix: sub_index_prop equivocating ix
Hnodup: NoDup
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
Hincl: incl
  (l ++
   [dec_exist 
      (sub_index_prop equivocating) ix Hix])
  (sub_index_listing equivocating)
i: index
aitems: list transition_item
afinal: _composite_state equivocator_IM
n: ¬ sub_index_prop equivocating i
IHl: afinal i = full_replay_state i
i ≠ ix
      congruence.
Qed.

For any equivocator_descriptors corresponding to the base state the projection of the replaying of an initial state is empty.

Lemma equivocators_trace_project_replayed_initial_state_from full_replay_state is
  (eqv_descriptors: equivocator_descriptors)
  (Heqv_descriptors: not_equivocating_equivocator_descriptors IM eqv_descriptors full_replay_state)
  : equivocators_trace_project eqv_descriptors
      (replayed_initial_state_from full_replay_state is) =
    Some ([], eqv_descriptors).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
equivocators_trace_project eqv_descriptors
  (replayed_initial_state_from full_replay_state is) =
Some ([], eqv_descriptors)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
equivocators_trace_project eqv_descriptors
  (replayed_initial_state_from full_replay_state is) =
Some ([], eqv_descriptors)
  unfold replayed_initial_state_from, spawn_initial_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
equivocators_trace_project eqv_descriptors
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map (initial_new_machine_transition_item is)
        (sub_index_listing equivocating))).1 =
Some ([], eqv_descriptors)
  generalize (@sub_index_listing index IndEqDec equivocating index_listing).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
∀ l : list (sub_index equivocating),
  equivocators_trace_project eqv_descriptors
    (composite_apply_plan equivocator_IM
       full_replay_state
       (map (initial_new_machine_transition_item is) l)).1 =
  Some ([], eqv_descriptors)
  intro l.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
l: list (sub_index equivocating)
equivocators_trace_project eqv_descriptors
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map (initial_new_machine_transition_item is) l)).1 =
Some ([], eqv_descriptors)
  remember (composite_apply_plan _ _ _) as plan.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
l: list (sub_index equivocating)
plan: (list transition_item * state)%type
Heqplan: plan =
composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l)
equivocators_trace_project eqv_descriptors plan.1 =
Some ([], eqv_descriptors)
  apply proj1 with (forall i, equivocator_state_n (full_replay_state i) <= equivocator_state_n (plan.2 i)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
l: list (sub_index equivocating)
plan: (list transition_item * state)%type
Heqplan: plan =
composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l)
equivocators_trace_project eqv_descriptors plan.1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (plan.2 i))
  subst plan.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
l: list (sub_index equivocating)
equivocators_trace_project eqv_descriptors
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map (initial_new_machine_transition_item is) l)).1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((composite_apply_plan equivocator_IM
             full_replay_state
             (map
                (initial_new_machine_transition_item
                   is) l)).2 i))
  induction l using rev_ind; [split; simpl; [reflexivity|lia]|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: equivocators_trace_project eqv_descriptors
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)).1 = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)).2 i))
equivocators_trace_project eqv_descriptors
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map (initial_new_machine_transition_item is)
        (l ++ [x]))).1 = Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((composite_apply_plan equivocator_IM
             full_replay_state
             (map
                (initial_new_machine_transition_item
                   is) (l ++ [x]))).2 i))
  rewrite map_app, (composite_apply_plan_app equivocator_IM).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: equivocators_trace_project eqv_descriptors
  (composite_apply_plan equivocator_IM
     full_replay_state
     (map
        (initial_new_machine_transition_item is)
        l)).1 = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)).2 i))
equivocators_trace_project eqv_descriptors
  (let (aitems, afinal) :=
     composite_apply_plan equivocator_IM
       full_replay_state
       (map (initial_new_machine_transition_item is) l) in
   let (a'items, a'final) :=
     composite_apply_plan equivocator_IM afinal
       (map (initial_new_machine_transition_item is)
          [x]) in
   (aitems ++ a'items, a'final)).1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((let (aitems, afinal) :=
             composite_apply_plan equivocator_IM
               full_replay_state
               (map
                  (initial_new_machine_transition_item
                     is) l) in
           let (a'items, a'final) :=
             composite_apply_plan equivocator_IM
               afinal
               (map
                  (initial_new_machine_transition_item
                     is) [x]) in
           (aitems ++ a'items, a'final)).2 i))
  destruct (composite_apply_plan _ _ _) as (litems, lfinal) eqn:Hplanl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: state
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  (litems, lfinal).1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((litems, lfinal).2 i))
equivocators_trace_project eqv_descriptors
  (let (a'items, a'final) :=
     composite_apply_plan equivocator_IM lfinal
       (map (initial_new_machine_transition_item is)
          [x]) in
   (litems ++ a'items, a'final)).1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((let (a'items, a'final) :=
             composite_apply_plan equivocator_IM
               lfinal
               (map
                  (initial_new_machine_transition_item
                     is) [x]) in
           (litems ++ a'items, a'final)).2 i))
  destruct (composite_apply_plan _ lfinal _) as (aitems, afinal) eqn:Hplana.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: state
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  (litems, lfinal).1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((litems, lfinal).2 i))
aitems: list transition_item
afinal: state
Hplana: composite_apply_plan equivocator_IM lfinal
  (map
     (initial_new_machine_transition_item is)
     [x]) = 
(aitems, afinal)
equivocators_trace_project eqv_descriptors
  (litems ++ aitems, afinal).1 =
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         ((litems ++ aitems, afinal).2 i))
  simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Hplana: composite_apply_plan equivocator_IM lfinal
  [initial_new_machine_transition_item is x] =
(aitems, afinal)
equivocators_trace_project eqv_descriptors
  (litems ++ aitems) = Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (afinal i))
  inversion_clear Hplana.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
equivocators_trace_project eqv_descriptors
  (litems ++
   [{|
      l :=
        existT (` x)
          (Spawn (equivocator_state_zero (is x)));
      input := None;
      destination :=
        state_update equivocator_IM lfinal (` x)
          (equivocator_state_extend (lfinal (` x))
             (equivocator_state_zero (is x)));
      output := None
    |}]) = Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n
         (state_update equivocator_IM lfinal (` x)
            (equivocator_state_extend (lfinal (` x))
               (equivocator_state_zero (is x))) i))
  split.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
equivocators_trace_project eqv_descriptors
  (litems ++
   [{|
      l :=
        existT (` x)
          (Spawn (equivocator_state_zero (is x)));
      input := None;
      destination :=
        state_update equivocator_IM lfinal (` x)
          (equivocator_state_extend (lfinal (` x))
             (equivocator_state_zero (is x)));
      output := None
    |}]) = Some ([], eqv_descriptors)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
∀ i : index,
  equivocator_state_n (full_replay_state i)
  ≤ equivocator_state_n
      (state_update equivocator_IM lfinal 
         (` x)
         (equivocator_state_extend 
         (lfinal (` x))
         (equivocator_state_zero (is x))) i)
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
equivocators_trace_project eqv_descriptors
  (litems ++
   [{|
      l :=
        existT (` x)
          (Spawn (equivocator_state_zero (is x)));
      input := None;
      destination :=
        state_update equivocator_IM lfinal (` x)
          (equivocator_state_extend (lfinal (` x))
             (equivocator_state_zero (is x)));
      output := None
    |}]) = Some ([], eqv_descriptors) apply equivocators_trace_project_app_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
∃ (preX sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : Common.equivocator_descriptors
                        IM),
  Projections.equivocators_trace_project IM
    eqv_descriptors
    [{|
       l :=
         existT (` x)
           (Spawn (equivocator_state_zero (is x)));
       input := None;
       destination :=
         state_update equivocator_IM lfinal (` x)
           (equivocator_state_extend (lfinal (` x))
              (equivocator_state_zero (is x)));
       output := None
     |}] = Some (sufX, eqv_descriptors')
  ∧ Projections.equivocators_trace_project IM
      eqv_descriptors' litems =
    Some (preX, eqv_descriptors) ∧ [] = preX ++ sufX
    exists [],[],eqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
∧ Projections.equivocators_trace_project IM
    eqv_descriptors litems =
  Some ([], eqv_descriptors) ∧ [] = [] ++ []
    repeat split; [|apply IHl].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    specialize (Heqv_descriptors (` x)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
Heqv_descriptors: existing_descriptor 
  (IM (` x))
  (eqv_descriptors (` x))
  (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    unfold existing_descriptor in Heqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
Heqv_descriptors: match eqv_descriptors (` x) with
| NewMachine _ => False
| Existing i =>
    is_Some
      (equivocator_state_project
         (full_replay_state (` x))
         i)
end
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    destruct (eqv_descriptors (` x)) eqn:Heqv_x; [contradiction|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
Heqv_descriptors: is_Some
  (equivocator_state_project
     (full_replay_state (` x)) n)
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    destruct Heqv_descriptors as [s_x_n Heqv_descriptors].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    apply equivocator_state_project_Some_rev in Heqv_descriptors as Hltn.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    apply proj2 in IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ i : index,
  equivocator_state_n (full_replay_state i)
  ≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    specialize (IHl (` x)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Projections.equivocators_trace_project IM
  eqv_descriptors
  [{|
     l :=
       existT (` x)
         (Spawn (equivocator_state_zero (is x)));
     input := None;
     destination :=
       state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x)));
     output := None
   |}] = Some ([], eqv_descriptors)
    cbn.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  equivocators_transition_item_project IM
    eqv_descriptors
    {|
      l :=
        existT (` x)
          (Spawn (equivocator_state_zero (is x)));
      input := None;
      destination :=
        state_update equivocator_IM lfinal (` x)
          (equivocator_state_extend (lfinal (` x))
             (equivocator_state_zero (is x)));
      output := None
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors) unfold equivocators_transition_item_project; simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  match
    equivocator_vlsm_transition_item_project
      (IM (` x))
      (composite_transition_item_projection
         (Common.equivocator_IM IM)
         {|
           l :=
             existT (` x)
               (Spawn (equivocator_state_zero (is x)));
           input := None;
           destination :=
             state_update equivocator_IM lfinal (` x)
               (equivocator_state_extend
                  (lfinal (` x))
                  (equivocator_state_zero (is x)));
           output := None
         |}) (eqv_descriptors (` x))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
    unfold equivocator_vlsm_transition_item_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  match
    match eqv_descriptors (` x) with
    | NewMachine _ =>
        Some (None, eqv_descriptors (` x))
    | Existing j =>
        match
          composite_transition_item_projection
            (Common.equivocator_IM IM)
            {|
              l :=
                existT (` x)
                  (Spawn
                     (equivocator_state_zero (is x)));
              input := None;
              destination :=
                state_update equivocator_IM lfinal
                  (` x)
                  (equivocator_state_extend
                     (lfinal (` x))
                     (equivocator_state_zero (is x)));
              output := None
            |}
        with
        | {|
            l := el;
            input := im;
            destination := s;
            output := om
          |} =>
            match equivocator_state_project s j with
            | Some sj =>
                match el with
                | Spawn sn =>
                    if
                     decide
                       (j = equivocator_state_last s)
                    then Some (None, NewMachine sn)
                    else Some (None, Existing j)
                | ContinueWith i lx =>
                    if decide (i = j)
                    then
                     Some
                       (Some
                          {|
                            l := lx;
                            input := im;
                            destination := sj;
                            output := om
                          |}, Existing i)
                    else Some (None, Existing j)
                | ForkWith i lx =>
                    if
                     decide
                       (j = equivocator_state_last s)
                    then
                     Some
                       (Some
                          {|
                            l := lx;
                            input := im;
                            destination := sj;
                            output := om
                          |}, Existing i)
                    else Some (None, Existing j)
                end
            | None => None
            end
        end
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors) rewrite Heqv_x.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  match
    match
      composite_transition_item_projection
        (Common.equivocator_IM IM)
        {|
          l :=
            existT (` x)
              (Spawn (equivocator_state_zero (is x)));
          input := None;
          destination :=
            state_update equivocator_IM lfinal (` x)
              (equivocator_state_extend (lfinal (` x))
                 (equivocator_state_zero (is x)));
          output := None
        |}
    with
    | {|
        l := el;
        input := im;
        destination := s;
        output := om
      |} =>
        match equivocator_state_project s n with
        | Some sj =>
            match el with
            | Spawn sn =>
                if
                 decide (n = equivocator_state_last s)
                then Some (None, NewMachine sn)
                else Some (None, Existing n)
            | ContinueWith i lx =>
                if decide (i = n)
                then
                 Some
                   (Some
                      {|
                        l := lx;
                        input := im;
                        destination := sj;
                        output := om
                      |}, Existing i)
                else Some (None, Existing n)
            | ForkWith i lx =>
                if
                 decide (n = equivocator_state_last s)
                then
                 Some
                   (Some
                      {|
                        l := lx;
                        input := im;
                        destination := sj;
                        output := om
                      |}, Existing i)
                else Some (None, Existing n)
            end
        | None => None
        end
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
    simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  match
    match
      equivocator_state_project
        (state_update equivocator_IM lfinal (` x)
           (equivocator_state_extend (lfinal (` x))
              (equivocator_state_zero (is x))) (` x))
        n
    with
    | Some _ =>
        if
         decide
           (n =
            equivocator_state_last
              (state_update equivocator_IM lfinal
                 (` x)
                 (equivocator_state_extend
                    (lfinal (` x))
                    (equivocator_state_zero (is x)))
                 (` x)))
        then
         Some
           (None,
           NewMachine (equivocator_state_zero (is x)))
        else Some (None, Existing n)
    | None => None
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors) rewrite state_update_eq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  match
    match
      equivocator_state_project
        (equivocator_state_extend (lfinal (` x))
           (equivocator_state_zero (is x))) n
    with
    | Some _ =>
        if
         decide
           (n =
            equivocator_state_last
              (equivocator_state_extend (lfinal (` x))
                 (equivocator_state_zero (is x))))
        then
         Some
           (None,
           NewMachine (equivocator_state_zero (is x)))
        else Some (None, Existing n)
    | None => None
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
    rewrite equivocator_state_extend_project_1 by lia.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
match
  match
    match
      equivocator_state_project (lfinal (` x)) n
    with
    | Some _ =>
        if
         decide
           (n =
            equivocator_state_last
              (equivocator_state_extend (lfinal (` x))
                 (equivocator_state_zero (is x))))
        then
         Some
           (None,
           NewMachine (equivocator_state_zero (is x)))
        else Some (None, Existing n)
    | None => None
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
    destruct_equivocator_state_project (lfinal (` x)) n lfinal_x_n Hltn'; [|lia].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
lfinal_x_n: vstate (IM (` x))
Hltn': n < equivocator_state_n (lfinal (` x))
match
  match
    (if
      decide
        (n =
         equivocator_state_last
           (equivocator_state_extend (lfinal (` x))
              (equivocator_state_zero (is x))))
     then
      Some
        (None,
        NewMachine (equivocator_state_zero (is x)))
     else Some (None, Existing n))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
    rewrite equivocator_state_extend_lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
lfinal_x_n: vstate (IM (` x))
Hltn': n < equivocator_state_n (lfinal (` x))
match
  match
    (if
      decide (n = equivocator_state_n (lfinal (` x)))
     then
      Some
        (None,
        NewMachine (equivocator_state_zero (is x)))
     else Some (None, Existing n))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (` x) (VLSM.l item');
             input := None;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (state_update equivocator_IM lfinal
                    (` x)
                    (equivocator_state_extend
                       (lfinal (` x))
                       (equivocator_state_zero (is x))));
             output := None
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors (` x) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
    rewrite decide_False by lia.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
lfinal_x_n: vstate (IM (` x))
Hltn': n < equivocator_state_n (lfinal (` x))
Some
  ([],
  Common.equivocator_descriptors_update IM
    eqv_descriptors (` x) (Existing n)) =
Some ([], eqv_descriptors)
    f_equal.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
lfinal_x_n: vstate (IM (` x))
Hltn': n < equivocator_state_n (lfinal (` x))
([],
Common.equivocator_descriptors_update IM
  eqv_descriptors (` x) (Existing n)) =
([], eqv_descriptors) f_equal.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
lfinal_x_n: vstate (IM (` x))
Hltn': n < equivocator_state_n (lfinal (` x))
Common.equivocator_descriptors_update IM
  eqv_descriptors (` x) (Existing n) = eqv_descriptors apply equivocator_descriptors_update_id.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
x: sub_index equivocating
n: nat
Heqv_x: eqv_descriptors (` x) = Existing n
s_x_n: vstate (IM (` x))
Heqv_descriptors: equivocator_state_project
  (full_replay_state (` x)) n =
Some s_x_n
Hltn: n <
equivocator_state_n (full_replay_state (` x))
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
lfinal_x_n: vstate (IM (` x))
Hltn': n < equivocator_state_n (lfinal (` x))
eqv_descriptors (` x) = Existing n assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
∀ i : index,
  equivocator_state_n (full_replay_state i)
  ≤ equivocator_state_n
      (state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x))) i) intro i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocators_trace_project eqv_descriptors
  litems = 
Some ([], eqv_descriptors)
∧ (∀ i : index,
     equivocator_state_n (full_replay_state i)
     ≤ equivocator_state_n (lfinal i))
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) i) apply proj2 in IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ i : index,
  equivocator_state_n (full_replay_state i)
  ≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) i) specialize (IHl i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
i: index
IHl: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) i)
    destruct (decide (` x = i)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
i: index
IHl: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
e: ` x = i
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
i: index
IHl: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
n: ` x ≠ i
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal 
       (` x)
       (equivocator_state_extend 
         (lfinal (` x))
         (equivocator_state_zero (is x))) i)
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
i: index
IHl: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
e: ` x = i
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) i) subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) (` x)) rewrite state_update_eq, equivocator_state_extend_size.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: equivocator_state_n (full_replay_state (` x))
≤ equivocator_state_n (lfinal (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
equivocator_state_n (full_replay_state (` x))
≤ S (equivocator_state_n (lfinal (` x))) lia.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
i: index
IHl: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
n: ` x ≠ i
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (state_update equivocator_IM lfinal (` x)
       (equivocator_state_extend (lfinal (` x))
          (equivocator_state_zero (is x))) i) rewrite state_update_neq by congruence.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
i: index
IHl: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i)
aitems: list transition_item
afinal: _composite_state equivocator_IM
n: ` x ≠ i
equivocator_state_n (full_replay_state i)
≤ equivocator_state_n (lfinal i) assumption.
Qed.

Lemma equivocator_state_project_replayed_initial_state_from_left full_replay_state is
  (lst := finite_trace_last full_replay_state (replayed_initial_state_from full_replay_state is))
  : forall i j,
    j < equivocator_state_n (full_replay_state i) ->
    equivocator_state_project (lst i) j =
    equivocator_state_project (full_replay_state i) j.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project (lst i) j =
    equivocator_state_project (full_replay_state i) j
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project (lst i) j =
    equivocator_state_project (full_replay_state i) j
  subst lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is) i) j =
    equivocator_state_project (full_replay_state i) j
  unfold replayed_initial_state_from, spawn_initial_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
            full_replay_state
            (map
               (initial_new_machine_transition_item is)
               (sub_index_listing equivocating))).1 i)
      j =
    equivocator_state_project (full_replay_state i) j
  generalize (@sub_index_listing index IndEqDec equivocating index_listing).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
∀ (l : list (sub_index equivocating)) (i : index) 
  (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
            full_replay_state
            (map
               (initial_new_machine_transition_item is)
               l)).1 i) j =
    equivocator_state_project (full_replay_state i) j
  intro l.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
l: list (sub_index equivocating)
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
            full_replay_state
            (map
               (initial_new_machine_transition_item is)
               l)).1 i) j =
    equivocator_state_project (full_replay_state i) j
  induction l using rev_ind; simpl; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)).1 i) j =
    equivocator_state_project
      (full_replay_state i) j
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
            full_replay_state
            (map
               (initial_new_machine_transition_item is)
               (l ++ [x]))).1 i) j =
    equivocator_state_project (full_replay_state i) j
  rewrite map_app, (composite_apply_plan_app equivocator_IM).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)).1 i) j =
    equivocator_state_project
      (full_replay_state i) j
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (let (aitems, afinal) :=
            composite_apply_plan equivocator_IM
              full_replay_state
              (map
                 (initial_new_machine_transition_item
                    is) l) in
          let (a'items, a'final) :=
            composite_apply_plan equivocator_IM afinal
              (map
                 (initial_new_machine_transition_item
                    is) [x]) in
          (aitems ++ a'items, a'final)).1 i) j =
    equivocator_state_project (full_replay_state i) j
  specialize (composite_apply_plan_last equivocator_IM full_replay_state (map (initial_new_machine_transition_item is) l))
    as Hlst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (composite_apply_plan equivocator_IM
         full_replay_state
         (map
         (initial_new_machine_transition_item is)
         l)).1 i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: let after_a :=
  composite_apply_plan equivocator_IM
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l) in
finite_trace_last full_replay_state after_a.1 =
after_a.2
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (let (aitems, afinal) :=
            composite_apply_plan equivocator_IM
              full_replay_state
              (map
                 (initial_new_machine_transition_item
                    is) l) in
          let (a'items, a'final) :=
            composite_apply_plan equivocator_IM afinal
              (map
                 (initial_new_machine_transition_item
                    is) [x]) in
          (aitems ++ a'items, a'final)).1 i) j =
    equivocator_state_project (full_replay_state i) j
  destruct (composite_apply_plan _ _ _) as (litems, lfinal) eqn:Hplanl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: state
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (litems, lfinal).1 i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: let after_a := (litems, lfinal) in
finite_trace_last full_replay_state after_a.1 =
after_a.2
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (let (a'items, a'final) :=
            composite_apply_plan equivocator_IM lfinal
              (map
                 (initial_new_machine_transition_item
                    is) [x]) in
          (litems ++ a'items, a'final)).1 i) j =
    equivocator_state_project (full_replay_state i) j
  destruct (composite_apply_plan _ lfinal _) as (aitems, afinal) eqn:Hplana.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: state
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (litems, lfinal).1 i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: let after_a := (litems, lfinal) in
finite_trace_last full_replay_state after_a.1 =
after_a.2
aitems: list transition_item
afinal: state
Hplana: composite_apply_plan equivocator_IM lfinal
  (map
     (initial_new_machine_transition_item is)
     [x]) = 
(aitems, afinal)
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (litems ++ aitems, afinal).1 i) j =
    equivocator_state_project (full_replay_state i) j
  inversion_clear Hplana.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: state
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (litems, lfinal).1 i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: let after_a := (litems, lfinal) in
finite_trace_last full_replay_state after_a.1 =
after_a.2
aitems: list transition_item
afinal: state
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (litems ++
          [{|
             l :=
               existT (` x)
                 (Spawn
                    (equivocator_state_zero (is x)));
             input := None;
             destination :=
               state_update equivocator_IM lfinal
                 (` x)
                 (equivocator_state_extend
                    (lfinal (` x))
                    (equivocator_state_zero (is x)));
             output := None
           |}],
         state_update equivocator_IM lfinal (` x)
           (equivocator_state_extend (lfinal (` x))
              (equivocator_state_zero (is x)))).1 i) j =
    equivocator_state_project (full_replay_state i) j
  simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: finite_trace_last full_replay_state litems =
lfinal
aitems: list transition_item
afinal: _composite_state equivocator_IM
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (litems ++
          [{|
             l :=
               existT (` x)
                 (Spawn
                    (equivocator_state_zero (is x)));
             input := None;
             destination :=
               state_update equivocator_IM lfinal
                 (` x)
                 (equivocator_state_extend
                    (lfinal (` x))
                    (equivocator_state_zero (is x)));
             output := None
           |}]) i) j =
    equivocator_state_project (full_replay_state i) j
  rewrite finite_trace_last_is_last.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: finite_trace_last full_replay_state litems =
lfinal
aitems: list transition_item
afinal: _composite_state equivocator_IM
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (destination
         {|
           l :=
             existT (` x)
               (Spawn (equivocator_state_zero (is x)));
           input := None;
           destination :=
             state_update equivocator_IM lfinal (` x)
               (equivocator_state_extend
                  (lfinal (` x))
                  (equivocator_state_zero (is x)));
           output := None
         |} i) j =
    equivocator_state_project (full_replay_state i) j simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: finite_trace_last full_replay_state litems =
lfinal
aitems: list transition_item
afinal: _composite_state equivocator_IM
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (state_update equivocator_IM lfinal (` x)
         (equivocator_state_extend (lfinal (` x))
            (equivocator_state_zero (is x))) i) j =
    equivocator_state_project (full_replay_state i) j
  intros i j Hj.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
lfinal: _composite_state equivocator_IM
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) = 
(litems, lfinal)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
Hlst: finite_trace_last full_replay_state litems =
lfinal
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
j: nat
Hj: j < equivocator_state_n (full_replay_state i)
equivocator_state_project
  (state_update equivocator_IM lfinal (` x)
     (equivocator_state_extend (lfinal (` x))
        (equivocator_state_zero (is x))) i) j =
equivocator_state_project (full_replay_state i) j
  destruct (decide (` x = i)); subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
j: nat
Hj: j < equivocator_state_n (full_replay_state (` x))
equivocator_state_project
  (state_update equivocator_IM
     (finite_trace_last full_replay_state litems)
     (` x)
     (equivocator_state_extend
        (finite_trace_last full_replay_state litems
           (` x)) (equivocator_state_zero (is x)))
     (` x)) j =
equivocator_state_project (full_replay_state (` x)) j
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
j: nat
Hj: j < equivocator_state_n (full_replay_state i)
n: ` x ≠ i
equivocator_state_project
  (state_update equivocator_IM
     (finite_trace_last full_replay_state litems)
     (` x)
     (equivocator_state_extend
        (finite_trace_last full_replay_state litems
         (` x)) 
        (equivocator_state_zero (is x))) i) j =
equivocator_state_project (full_replay_state i) j
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
j: nat
Hj: j < equivocator_state_n (full_replay_state (` x))
equivocator_state_project
  (state_update equivocator_IM
     (finite_trace_last full_replay_state litems)
     (` x)
     (equivocator_state_extend
        (finite_trace_last full_replay_state litems
           (` x)) (equivocator_state_zero (is x)))
     (` x)) j =
equivocator_state_project (full_replay_state (` x)) j rewrite state_update_eq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
j: nat
Hj: j < equivocator_state_n (full_replay_state (` x))
equivocator_state_project
  (equivocator_state_extend
     (finite_trace_last full_replay_state litems (` x))
     (equivocator_state_zero (is x))) j =
equivocator_state_project (full_replay_state (` x)) j
    specialize (IHl (` x) j Hj).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
j: nat
IHl: equivocator_state_project
  (finite_trace_last full_replay_state litems
     (` x)) j =
equivocator_state_project
  (full_replay_state (` x)) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
Hj: j < equivocator_state_n (full_replay_state (` x))
equivocator_state_project
  (equivocator_state_extend
     (finite_trace_last full_replay_state litems (` x))
     (equivocator_state_zero (is x))) j =
equivocator_state_project (full_replay_state (` x)) j
    destruct_equivocator_state_project (full_replay_state (` x)) j s_x_j Hltj; [|lia].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
j: nat
s_x_j: vstate (IM (` x))
IHl: equivocator_state_project
  (finite_trace_last full_replay_state litems
     (` x)) j = 
Some s_x_j
aitems: list transition_item
afinal: _composite_state equivocator_IM
Hj, Hltj: j <
equivocator_state_n (full_replay_state (` x))
equivocator_state_project
  (equivocator_state_extend
     (finite_trace_last full_replay_state litems (` x))
     (equivocator_state_zero (is x))) j = Some s_x_j
    rewrite equivocator_state_extend_project_1; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
j: nat
s_x_j: vstate (IM (` x))
IHl: equivocator_state_project
  (finite_trace_last full_replay_state litems
     (` x)) j = 
Some s_x_j
aitems: list transition_item
afinal: _composite_state equivocator_IM
Hj, Hltj: j <
equivocator_state_n (full_replay_state (` x))
j <
equivocator_state_n
  (finite_trace_last full_replay_state litems (` x))
    apply equivocator_state_project_Some_rev in IHl as Hltj'.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
j: nat
s_x_j: vstate (IM (` x))
IHl: equivocator_state_project
  (finite_trace_last full_replay_state litems
     (` x)) j = 
Some s_x_j
Hltj': j <
equivocator_state_n
  (finite_trace_last full_replay_state litems
     (` x))
aitems: list transition_item
afinal: _composite_state equivocator_IM
Hj, Hltj: j <
equivocator_state_n (full_replay_state (` x))
j <
equivocator_state_n
  (finite_trace_last full_replay_state litems (` x))
    assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
j: nat
Hj: j < equivocator_state_n (full_replay_state i)
n: ` x ≠ i
equivocator_state_project
  (state_update equivocator_IM
     (finite_trace_last full_replay_state litems)
     (` x)
     (equivocator_state_extend
        (finite_trace_last full_replay_state litems
           (` x)) (equivocator_state_zero (is x))) i)
  j =
equivocator_state_project (full_replay_state i) j rewrite state_update_neq by congruence.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
j: nat
Hj: j < equivocator_state_n (full_replay_state i)
n: ` x ≠ i
equivocator_state_project
  (finite_trace_last full_replay_state litems i) j =
equivocator_state_project (full_replay_state i) j
    apply IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: _composite_state equivocator_IM
is: composite_state sub_equivocator_IM
x: sub_index equivocating
l: list (sub_index equivocating)
litems: list transition_item
Hplanl: composite_apply_plan equivocator_IM
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     l) =
(litems,
finite_trace_last full_replay_state litems)
IHl: ∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         litems i) j =
    equivocator_state_project
      (full_replay_state i) j
aitems: list transition_item
afinal: _composite_state equivocator_IM
i: index
j: nat
Hj: j < equivocator_state_n (full_replay_state i)
n: ` x ≠ i
j < equivocator_state_n (full_replay_state i)
    by assumption.
Qed.

Lemma equivocator_state_descriptor_project_replayed_initial_state_from_left full_replay_state is
  (eqv_descriptors: equivocator_descriptors)
  (Heqv_descriptors: not_equivocating_equivocator_descriptors IM eqv_descriptors full_replay_state)
  (lst := finite_trace_last full_replay_state (replayed_initial_state_from full_replay_state is))
  : forall i,
    equivocator_state_descriptor_project (lst i) (eqv_descriptors i) =
    equivocator_state_descriptor_project (full_replay_state i) (eqv_descriptors i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
∀ i : index,
  equivocator_state_descriptor_project (lst i)
    (eqv_descriptors i) =
  equivocator_state_descriptor_project
    (full_replay_state i) (eqv_descriptors i)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
∀ i : index,
  equivocator_state_descriptor_project (lst i)
    (eqv_descriptors i) =
  equivocator_state_descriptor_project
    (full_replay_state i) (eqv_descriptors i)
  intro i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
equivocator_state_descriptor_project (lst i)
  (eqv_descriptors i) =
equivocator_state_descriptor_project
  (full_replay_state i) (eqv_descriptors i) spec Heqv_descriptors i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
Heqv_descriptors: existing_descriptor 
  (IM i) (eqv_descriptors i)
  (full_replay_state i)
equivocator_state_descriptor_project (lst i)
  (eqv_descriptors i) =
equivocator_state_descriptor_project
  (full_replay_state i) (eqv_descriptors i)
  unfold equivocator_state_descriptor_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
Heqv_descriptors: existing_descriptor 
  (IM i) (eqv_descriptors i)
  (full_replay_state i)
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default (equivocator_state_zero (lst i))
      (equivocator_state_project (lst i) j)
end =
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default
      (equivocator_state_zero (full_replay_state i))
      (equivocator_state_project (full_replay_state i)
         j)
end
  unfold existing_descriptor in Heqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
Heqv_descriptors: match eqv_descriptors i with
| NewMachine _ => False
| Existing i0 =>
    is_Some
      (equivocator_state_project
         (full_replay_state i) i0)
end
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default (equivocator_state_zero (lst i))
      (equivocator_state_project (lst i) j)
end =
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default
      (equivocator_state_zero (full_replay_state i))
      (equivocator_state_project (full_replay_state i)
         j)
end
  destruct (eqv_descriptors i) as [sn|ji]; [contradiction|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
ji: nat
Heqv_descriptors: is_Some
  (equivocator_state_project
     (full_replay_state i) ji)
default (equivocator_state_zero (lst i))
  (equivocator_state_project (lst i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  destruct Heqv_descriptors as [full_i_ji Hpr_ji].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
default (equivocator_state_zero (lst i))
  (equivocator_state_project (lst i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  apply equivocator_state_project_Some_rev in Hpr_ji as Hltji.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state
     is): state
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default (equivocator_state_zero (lst i))
  (equivocator_state_project (lst i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  subst lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default
  (equivocator_state_zero
     (finite_trace_last full_replay_state
        (replayed_initial_state_from full_replay_state
           is) i))
  (equivocator_state_project
     (finite_trace_last full_replay_state
        (replayed_initial_state_from full_replay_state
           is) i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  rewrite equivocator_state_project_replayed_initial_state_from_left by assumption.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default
  (equivocator_state_zero
     (finite_trace_last full_replay_state
        (replayed_initial_state_from full_replay_state
           is) i))
  (equivocator_state_project (full_replay_state i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  rewrite Hpr_ji.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
eqv_descriptors: equivocator_descriptors
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default
  (equivocator_state_zero
     (finite_trace_last full_replay_state
        (replayed_initial_state_from full_replay_state
           is) i)) (Some full_i_ji) =
default (equivocator_state_zero (full_replay_state i))
  (Some full_i_ji)
  reflexivity.
Qed.

Definition replayed_trace_from full_replay_state is tr :=
  replayed_initial_state_from full_replay_state is ++
  pre_VLSM_full_projection_finite_trace_project (type FreeSubE) (type FreeE)
    (lift_equivocators_sub_label_to full_replay_state)
    (lift_equivocators_sub_state_to full_replay_state) tr.

Lemma replayed_trace_from_finite_trace_last full_replay_state is tr
  (His : composite_initial_state_prop _ is)
  : finite_trace_last full_replay_state (replayed_trace_from full_replay_state is tr) =
    (lift_equivocators_sub_state_to full_replay_state (finite_trace_last is tr)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is tr)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is tr)
  destruct_list_last tr tr' item Htr; subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is []) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is [])
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
tr': list transition_item
item: transition_item
finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is
     (tr' ++ [item])) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is (tr' ++ [item]))
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is []) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is []) unfold replayed_trace_from.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is ++
   pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) (type FreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     []) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is []) cbn.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is ++
   []) =
lift_equivocators_sub_state_to full_replay_state is
    rewrite app_nil_r.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is) =
lift_equivocators_sub_state_to full_replay_state is
    apply replayed_initial_state_from_lift.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
composite_initial_state_prop sub_equivocator_IM is
    assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
tr': list transition_item
item: transition_item
finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is
     (tr' ++ [item])) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is (tr' ++ [item])) unfold replayed_trace_from, pre_VLSM_full_projection_finite_trace_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
tr': list transition_item
item: transition_item
finite_trace_last full_replay_state
  (replayed_initial_state_from full_replay_state is ++
   map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) (tr' ++ [item])) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is (tr' ++ [item]))
    rewrite map_app, app_assoc.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
tr': list transition_item
item: transition_item
finite_trace_last full_replay_state
  ((replayed_initial_state_from full_replay_state is ++
    map
      (pre_VLSM_full_projection_trace_item_project
         (type FreeSubE) (type FreeE)
         (lift_equivocators_sub_label_to
            full_replay_state)
         (lift_equivocators_sub_state_to
            full_replay_state)) tr') ++
   map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) [item]) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is (tr' ++ [item])) simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
tr': list transition_item
item: transition_item
finite_trace_last full_replay_state
  ((replayed_initial_state_from full_replay_state is ++
    map
      (pre_VLSM_full_projection_trace_item_project
         (composite_type sub_equivocator_IM)
         (composite_type equivocator_IM)
         (lift_equivocators_sub_label_to
            full_replay_state)
         (lift_equivocators_sub_state_to
            full_replay_state)) tr') ++
   [pre_VLSM_full_projection_trace_item_project
      (composite_type sub_equivocator_IM)
      (composite_type equivocator_IM)
      (lift_equivocators_sub_label_to
         full_replay_state)
      (lift_equivocators_sub_state_to
         full_replay_state) item]) =
lift_equivocators_sub_state_to full_replay_state
  (finite_trace_last is (tr' ++ [item]))
    rewrite !finite_trace_last_is_last.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
tr': list transition_item
item: transition_item
destination
  (pre_VLSM_full_projection_trace_item_project
     (composite_type sub_equivocator_IM)
     (composite_type equivocator_IM)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     item) =
lift_equivocators_sub_state_to full_replay_state
  (destination item)
    reflexivity.
Qed.

Lemma equivocator_state_project_replayed_trace_from_left full_replay_state is tr
  (lst := finite_trace_last full_replay_state (replayed_trace_from full_replay_state is tr))
  : forall i j,
    j < equivocator_state_n (full_replay_state i) ->
    equivocator_state_project (lst i) j =
    equivocator_state_project (full_replay_state i) j.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project (lst i) j =
    equivocator_state_project (full_replay_state i) j
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project (lst i) j =
    equivocator_state_project (full_replay_state i) j
  subst lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_trace_from full_replay_state is tr)
         i) j =
    equivocator_state_project (full_replay_state i) j
  unfold replayed_trace_from.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is ++
          pre_VLSM_full_projection_finite_trace_project
            (type FreeSubE) (type FreeE)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state) tr) i) j =
    equivocator_state_project (full_replay_state i) j
  destruct_list_last tr tr' lst Htr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
Htr: tr = []
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is ++
          pre_VLSM_full_projection_finite_trace_project
            (type FreeSubE) (type FreeE)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state) []) i) j =
    equivocator_state_project (full_replay_state i) j
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
         full_replay_state is ++
         pre_VLSM_full_projection_finite_trace_project
         (type FreeSubE) 
         (type FreeE)
         (lift_equivocators_sub_label_to
         full_replay_state)
         (lift_equivocators_sub_state_to
         full_replay_state) 
         (tr' ++ [lst])) i) j =
    equivocator_state_project (full_replay_state i) j
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
Htr: tr = []
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is ++
          pre_VLSM_full_projection_finite_trace_project
            (type FreeSubE) (type FreeE)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state) []) i) j =
    equivocator_state_project (full_replay_state i) j simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
Htr: tr = []
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is ++ []) i) j =
    equivocator_state_project (full_replay_state i) j rewrite app_nil_r.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
Htr: tr = []
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is) i) j =
    equivocator_state_project (full_replay_state i) j
    apply equivocator_state_project_replayed_initial_state_from_left.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is ++
          pre_VLSM_full_projection_finite_trace_project
            (type FreeSubE) (type FreeE)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state) (tr' ++ [lst])) i) j =
    equivocator_state_project (full_replay_state i) j unfold pre_VLSM_full_projection_finite_trace_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         (replayed_initial_state_from
            full_replay_state is ++
          map
            (pre_VLSM_full_projection_trace_item_project
               (type FreeSubE) (type FreeE)
               (lift_equivocators_sub_label_to
                  full_replay_state)
               (lift_equivocators_sub_state_to
                  full_replay_state)) (tr' ++ [lst]))
         i) j =
    equivocator_state_project (full_replay_state i) j
    rewrite map_app, app_assoc.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         ((replayed_initial_state_from
             full_replay_state is ++
           map
             (pre_VLSM_full_projection_trace_item_project
                (type FreeSubE) (type FreeE)
                (lift_equivocators_sub_label_to
                   full_replay_state)
                (lift_equivocators_sub_state_to
                   full_replay_state)) tr') ++
          map
            (pre_VLSM_full_projection_trace_item_project
               (type FreeSubE) (type FreeE)
               (lift_equivocators_sub_label_to
                  full_replay_state)
               (lift_equivocators_sub_state_to
                  full_replay_state)) [lst]) i) j =
    equivocator_state_project (full_replay_state i) j simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (finite_trace_last full_replay_state
         ((replayed_initial_state_from
             full_replay_state is ++
           map
             (pre_VLSM_full_projection_trace_item_project
                (composite_type sub_equivocator_IM)
                (composite_type equivocator_IM)
                (lift_equivocators_sub_label_to
                   full_replay_state)
                (lift_equivocators_sub_state_to
                   full_replay_state)) tr') ++
          [pre_VLSM_full_projection_trace_item_project
             (composite_type sub_equivocator_IM)
             (composite_type equivocator_IM)
             (lift_equivocators_sub_label_to
                full_replay_state)
             (lift_equivocators_sub_state_to
                full_replay_state) lst]) i) j =
    equivocator_state_project (full_replay_state i) j
    rewrite finite_trace_last_is_last.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (destination
         (pre_VLSM_full_projection_trace_item_project
            (composite_type sub_equivocator_IM)
            (composite_type equivocator_IM)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state) lst) i) j =
    equivocator_state_project (full_replay_state i) j
    simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
∀ (i : index) (j : nat),
  j < equivocator_state_n (full_replay_state i)
  → equivocator_state_project
      (lift_equivocators_sub_state_to
         full_replay_state (destination lst) i) j =
    equivocator_state_project (full_replay_state i) j
    intros i j Hltj.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
i: index
j: nat
Hltj: j < equivocator_state_n (full_replay_state i)
equivocator_state_project
  (lift_equivocators_sub_state_to full_replay_state
     (destination lst) i) j =
equivocator_state_project (full_replay_state i) j
    unfold lift_equivocators_sub_state_to.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
i: index
j: nat
Hltj: j < equivocator_state_n (full_replay_state i)
equivocator_state_project
  match decide (sub_index_prop equivocating i) with
  | left e =>
      equivocator_state_append (full_replay_state i)
        (destination lst
           (dec_exist (sub_index_prop equivocating) i
              e))
  | in_right => full_replay_state i
  end j =
equivocator_state_project (full_replay_state i) j
    destruct (decide _); [|reflexivity].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
i: index
j: nat
Hltj: j < equivocator_state_n (full_replay_state i)
s: sub_index_prop equivocating i
equivocator_state_project
  (equivocator_state_append (full_replay_state i)
     (destination lst
        (dec_exist (sub_index_prop equivocating) i s)))
  j =
equivocator_state_project (full_replay_state i) j
    rewrite equivocator_state_append_project_1 by assumption.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr, tr': list transition_item
lst: transition_item
Htr: tr = tr' ++ [lst]
i: index
j: nat
Hltj: j < equivocator_state_n (full_replay_state i)
s: sub_index_prop equivocating i
equivocator_state_project (full_replay_state i) j =
equivocator_state_project (full_replay_state i) j
    reflexivity.
Qed.

Lemma equivocator_state_descriptor_project_replayed_trace_from_left full_replay_state is tr
  (eqv_descriptors: equivocator_descriptors)
  (Heqv_descriptors: not_equivocating_equivocator_descriptors IM eqv_descriptors full_replay_state)
  (lst := finite_trace_last full_replay_state (replayed_trace_from full_replay_state is tr))
  : forall i,
    equivocator_state_descriptor_project (lst i) (eqv_descriptors i) =
    equivocator_state_descriptor_project (full_replay_state i) (eqv_descriptors i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
∀ i : index,
  equivocator_state_descriptor_project (lst i)
    (eqv_descriptors i) =
  equivocator_state_descriptor_project
    (full_replay_state i) (eqv_descriptors i)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
∀ i : index,
  equivocator_state_descriptor_project (lst i)
    (eqv_descriptors i) =
  equivocator_state_descriptor_project
    (full_replay_state i) (eqv_descriptors i)
  intro i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
equivocator_state_descriptor_project (lst i)
  (eqv_descriptors i) =
equivocator_state_descriptor_project
  (full_replay_state i) (eqv_descriptors i) spec Heqv_descriptors i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
Heqv_descriptors: existing_descriptor 
  (IM i) (eqv_descriptors i)
  (full_replay_state i)
equivocator_state_descriptor_project (lst i)
  (eqv_descriptors i) =
equivocator_state_descriptor_project
  (full_replay_state i) (eqv_descriptors i)
  unfold equivocator_state_descriptor_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
Heqv_descriptors: existing_descriptor 
  (IM i) (eqv_descriptors i)
  (full_replay_state i)
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default (equivocator_state_zero (lst i))
      (equivocator_state_project (lst i) j)
end =
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default
      (equivocator_state_zero (full_replay_state i))
      (equivocator_state_project (full_replay_state i)
         j)
end
  unfold existing_descriptor in Heqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
Heqv_descriptors: match eqv_descriptors i with
| NewMachine _ => False
| Existing i0 =>
    is_Some
      (equivocator_state_project
         (full_replay_state i) i0)
end
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default (equivocator_state_zero (lst i))
      (equivocator_state_project (lst i) j)
end =
match eqv_descriptors i with
| NewMachine sn => sn
| Existing j =>
    default
      (equivocator_state_zero (full_replay_state i))
      (equivocator_state_project (full_replay_state i)
         j)
end
  destruct (eqv_descriptors i) as [sn|ji]; [contradiction|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
ji: nat
Heqv_descriptors: is_Some
  (equivocator_state_project
     (full_replay_state i) ji)
default (equivocator_state_zero (lst i))
  (equivocator_state_project (lst i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  destruct Heqv_descriptors as [full_i_ji Hpr_ji].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
default (equivocator_state_zero (lst i))
  (equivocator_state_project (lst i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  apply equivocator_state_project_Some_rev in Hpr_ji as Hltji.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default (equivocator_state_zero (lst i))
  (equivocator_state_project (lst i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  subst lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default
  (equivocator_state_zero
     (finite_trace_last full_replay_state
        (replayed_trace_from full_replay_state is tr)
        i))
  (equivocator_state_project
     (finite_trace_last full_replay_state
        (replayed_trace_from full_replay_state is tr)
        i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  rewrite equivocator_state_project_replayed_trace_from_left by assumption.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default
  (equivocator_state_zero
     (finite_trace_last full_replay_state
        (replayed_trace_from full_replay_state is tr)
        i))
  (equivocator_state_project (full_replay_state i) ji) =
default (equivocator_state_zero (full_replay_state i))
  (equivocator_state_project (full_replay_state i) ji)
  rewrite Hpr_ji.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
i: index
ji: nat
full_i_ji: vstate (IM i)
Hpr_ji: equivocator_state_project
  (full_replay_state i) ji = 
Some full_i_ji
Hltji: ji < equivocator_state_n (full_replay_state i)
default
  (equivocator_state_zero
     (finite_trace_last full_replay_state
        (replayed_trace_from full_replay_state is tr)
        i)) (Some full_i_ji) =
default (equivocator_state_zero (full_replay_state i))
  (Some full_i_ji)
  reflexivity.
Qed.

Lemma equivocators_total_state_project_replayed_trace_from full_replay_state is tr
  (lst := finite_trace_last full_replay_state (replayed_trace_from full_replay_state is tr))
  : equivocators_total_state_project IM lst = equivocators_total_state_project IM full_replay_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
equivocators_total_state_project IM lst =
equivocators_total_state_project IM full_replay_state
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
equivocators_total_state_project IM lst =
equivocators_total_state_project IM full_replay_state
  apply functional_extensionality_dep.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
∀ x : index,
  equivocators_total_state_project IM lst x =
  equivocators_total_state_project IM
    full_replay_state x
  intro i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
equivocators_total_state_project IM lst i =
equivocators_total_state_project IM full_replay_state
  i
  apply equivocator_state_descriptor_project_replayed_trace_from_left.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
lst:= finite_trace_last full_replay_state
  (replayed_trace_from full_replay_state is tr): state
i: index
not_equivocating_equivocator_descriptors IM
  (zero_descriptor IM) full_replay_state
  apply zero_descriptor_not_equivocating.
Qed.

Lemma equivocators_trace_project_replayed_trace_from_left full_replay_state is tr
  (eqv_descriptors: equivocator_descriptors)
  (Heqv_descriptors: not_equivocating_equivocator_descriptors IM eqv_descriptors full_replay_state)
  : equivocators_trace_project eqv_descriptors
      (replayed_trace_from full_replay_state is tr) =
    Some ([], eqv_descriptors).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
equivocators_trace_project eqv_descriptors
  (replayed_trace_from full_replay_state is tr) =
Some ([], eqv_descriptors)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
equivocators_trace_project eqv_descriptors
  (replayed_trace_from full_replay_state is tr) =
Some ([], eqv_descriptors)
  apply equivocators_trace_project_app_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
∃ (preX sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : Common.equivocator_descriptors
                        IM),
  Projections.equivocators_trace_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_finite_trace_project
       (type FreeSubE) (type FreeE)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state) tr) =
  Some (sufX, eqv_descriptors')
  ∧ Projections.equivocators_trace_project IM
      eqv_descriptors'
      (replayed_initial_state_from full_replay_state
         is) = Some (preX, eqv_descriptors)
    ∧ [] = preX ++ sufX
  exists [],[],eqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) (type FreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     tr) = Some ([], eqv_descriptors)
∧ Projections.equivocators_trace_project IM
    eqv_descriptors
    (replayed_initial_state_from full_replay_state is) =
  Some ([], eqv_descriptors) ∧ [] = [] ++ []
  repeat split; [|apply equivocators_trace_project_replayed_initial_state_from; assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) (type FreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     tr) = Some ([], eqv_descriptors)
  induction tr using rev_ind; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
IHtr: Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) 
     (type FreeE)
     (lift_equivocators_sub_label_to
        full_replay_state)
     (lift_equivocators_sub_state_to
        full_replay_state) tr) =
Some ([], eqv_descriptors)
Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) (type FreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     (tr ++ [x])) = Some ([], eqv_descriptors)
  unfold pre_VLSM_full_projection_finite_trace_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
IHtr: Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) 
     (type FreeE)
     (lift_equivocators_sub_label_to
        full_replay_state)
     (lift_equivocators_sub_state_to
        full_replay_state) tr) =
Some ([], eqv_descriptors)
Projections.equivocators_trace_project IM
  eqv_descriptors
  (map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) (tr ++ [x])) =
Some ([], eqv_descriptors)
  rewrite map_app.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
IHtr: Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) 
     (type FreeE)
     (lift_equivocators_sub_label_to
        full_replay_state)
     (lift_equivocators_sub_state_to
        full_replay_state) tr) =
Some ([], eqv_descriptors)
Projections.equivocators_trace_project IM
  eqv_descriptors
  (map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) tr ++
   map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) [x]) =
Some ([], eqv_descriptors)
  apply equivocators_trace_project_app_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
IHtr: Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) 
     (type FreeE)
     (lift_equivocators_sub_label_to
        full_replay_state)
     (lift_equivocators_sub_state_to
        full_replay_state) tr) =
Some ([], eqv_descriptors)
∃ (preX sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : Common.equivocator_descriptors
                        IM),
  Projections.equivocators_trace_project IM
    eqv_descriptors
    (map
       (pre_VLSM_full_projection_trace_item_project
          (type FreeSubE) (type FreeE)
          (lift_equivocators_sub_label_to
             full_replay_state)
          (lift_equivocators_sub_state_to
             full_replay_state)) [x]) =
  Some (sufX, eqv_descriptors')
  ∧ Projections.equivocators_trace_project IM
      eqv_descriptors'
      (map
         (pre_VLSM_full_projection_trace_item_project
            (type FreeSubE) (type FreeE)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state)) tr) =
    Some (preX, eqv_descriptors) ∧ [] = preX ++ sufX
  exists [],[], eqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
IHtr: Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) 
     (type FreeE)
     (lift_equivocators_sub_label_to
        full_replay_state)
     (lift_equivocators_sub_state_to
        full_replay_state) tr) =
Some ([], eqv_descriptors)
Projections.equivocators_trace_project IM
  eqv_descriptors
  (map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) [x]) =
Some ([], eqv_descriptors)
∧ Projections.equivocators_trace_project IM
    eqv_descriptors
    (map
       (pre_VLSM_full_projection_trace_item_project
          (type FreeSubE) (type FreeE)
          (lift_equivocators_sub_label_to
             full_replay_state)
          (lift_equivocators_sub_state_to
             full_replay_state)) tr) =
  Some ([], eqv_descriptors) ∧ [] = [] ++ []
  repeat split; [|assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
IHtr: Projections.equivocators_trace_project IM
  eqv_descriptors
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) 
     (type FreeE)
     (lift_equivocators_sub_label_to
        full_replay_state)
     (lift_equivocators_sub_state_to
        full_replay_state) tr) =
Some ([], eqv_descriptors)
Projections.equivocators_trace_project IM
  eqv_descriptors
  (map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) [x]) =
Some ([], eqv_descriptors)
  clear IHtr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
x: transition_item
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
Projections.equivocators_trace_project IM
  eqv_descriptors
  (map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state)) [x]) =
Some ([], eqv_descriptors)
  destruct x.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
l: label
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
Projections.equivocators_trace_project IM
  eqv_descriptors
  (map
     (pre_VLSM_full_projection_trace_item_project
        (type FreeSubE) (type FreeE)
        (lift_equivocators_sub_label_to
           full_replay_state)
        (lift_equivocators_sub_state_to
           full_replay_state))
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) = Some ([], eqv_descriptors) simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
l: label
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l := l;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  destruct l as (sub_i,li).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
sub_i: sub_index equivocating
li: vlabel (sub_equivocator_IM sub_i)
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l := existT sub_i li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  destruct_dec_sig sub_i i Hi Heqsub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
sub_i: sub_index equivocating
li: vlabel (sub_equivocator_IM sub_i)
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l := existT sub_i li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  subst sub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors
  full_replay_state
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  specialize (Heqv_descriptors i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: existing_descriptor 
  (IM i) (eqv_descriptors i)
  (full_replay_state i)
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  unfold existing_descriptor in Heqv_descriptors.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
Heqv_descriptors: match eqv_descriptors i with
| NewMachine _ => False
| Existing i0 =>
    is_Some
      (equivocator_state_project
         (full_replay_state i) i0)
end
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  destruct (eqv_descriptors _) eqn:Heqv_l; [contradiction|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
Heqv_descriptors: is_Some
  (equivocator_state_project
     (full_replay_state i) n)
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  destruct Heqv_descriptors as [s_l_n Hs_l_n].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  apply equivocator_state_project_Some_rev in Hs_l_n as Hltn.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  specialize (lift_equivocators_sub_state_to_size full_replay_state destination i)
    as Hltsize.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
match
  equivocators_transition_item_project IM
    eqv_descriptors
    (pre_VLSM_full_projection_trace_item_project
       (composite_type sub_equivocator_IM)
       (composite_type equivocator_IM)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state)
       {|
         l :=
           existT
             (dec_exist (sub_index_prop equivocating)
                i Hi) li;
         input := input;
         destination := destination;
         output := output
       |})
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  unfold equivocators_transition_item_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
match
  match
    equivocator_vlsm_transition_item_project
      (IM
         (projT1
            (l
               (pre_VLSM_full_projection_trace_item_project
                  (composite_type sub_equivocator_IM)
                  (composite_type equivocator_IM)
                  (lift_equivocators_sub_label_to
                     full_replay_state)
                  (lift_equivocators_sub_state_to
                     full_replay_state)
                  {|
                    l :=
                      existT
                        (dec_exist
                           (sub_index_prop
                              equivocating) i Hi) li;
                    input := input;
                    destination := destination;
                    output := output
                  |}))))
      (composite_transition_item_projection
         (Common.equivocator_IM IM)
         (pre_VLSM_full_projection_trace_item_project
            (composite_type sub_equivocator_IM)
            (composite_type equivocator_IM)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state)
            {|
              l :=
                existT
                  (dec_exist
                     (sub_index_prop equivocating) i
                     Hi) li;
              input := input;
              destination := destination;
              output := output
            |}))
      (eqv_descriptors
         (projT1
            (l
               (pre_VLSM_full_projection_trace_item_project
                  (composite_type sub_equivocator_IM)
                  (composite_type equivocator_IM)
                  (lift_equivocators_sub_label_to
                     full_replay_state)
                  (lift_equivocators_sub_state_to
                     full_replay_state)
                  {|
                    l :=
                      existT
                        (dec_exist
                           (sub_index_prop
                              equivocating) i Hi) li;
                    input := input;
                    destination := destination;
                    output := output
                  |}))))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l :=
               existT
                 (projT1
                    (l
                       (pre_VLSM_full_projection_trace_item_project
                          (composite_type
                             sub_equivocator_IM)
                          (composite_type
                             equivocator_IM)
                          (lift_equivocators_sub_label_to
                             full_replay_state)
                          (lift_equivocators_sub_state_to
                             full_replay_state)
                          {|
                            l :=
                              existT
                                (dec_exist (...) i Hi)
                                li;
                            input := input;
                            destination := destination;
                            output := output
                          |}))) (l item');
             input :=
               VLSM.input
                 (pre_VLSM_full_projection_trace_item_project
                    (composite_type sub_equivocator_IM)
                    (composite_type equivocator_IM)
                    (lift_equivocators_sub_label_to
                       full_replay_state)
                    (lift_equivocators_sub_state_to
                       full_replay_state)
                    {|
                      l :=
                        existT
                          (dec_exist
                             (sub_index_prop
                                equivocating) i Hi) li;
                      input := input;
                      destination := destination;
                      output := output
                    |});
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (VLSM.destination
                    (pre_VLSM_full_projection_trace_item_project
                       (composite_type
                          sub_equivocator_IM)
                       (composite_type equivocator_IM)
                       (lift_equivocators_sub_label_to
                          full_replay_state)
                       (lift_equivocators_sub_state_to
                          full_replay_state)
                       {|
                         l :=
                           existT
                             (dec_exist
                                (sub_index_prop
                                   equivocating) i Hi)
                             li;
                         input := input;
                         destination := destination;
                         output := output
                       |}));
             output :=
               VLSM.output
                 (pre_VLSM_full_projection_trace_item_project
                    (composite_type sub_equivocator_IM)
                    (composite_type equivocator_IM)
                    (lift_equivocators_sub_label_to
                       full_replay_state)
                    (lift_equivocators_sub_state_to
                       full_replay_state)
                    {|
                      l :=
                        existT
                          (dec_exist
                             (sub_index_prop
                                equivocating) i Hi) li;
                      input := input;
                      destination := destination;
                      output := output
                    |})
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors
          (projT1
             (l
                (pre_VLSM_full_projection_trace_item_project
                   (composite_type sub_equivocator_IM)
                   (composite_type equivocator_IM)
                   (lift_equivocators_sub_label_to
                      full_replay_state)
                   (lift_equivocators_sub_state_to
                      full_replay_state)
                   {|
                     l :=
                       existT
                         (dec_exist
                            (sub_index_prop
                               equivocating) i Hi) li;
                     input := input;
                     destination := destination;
                     output := output
                   |}))) deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors
          (projT1
             (l
                (pre_VLSM_full_projection_trace_item_project
                   (composite_type sub_equivocator_IM)
                   (composite_type equivocator_IM)
                   (lift_equivocators_sub_label_to
                      full_replay_state)
                   (lift_equivocators_sub_state_to
                      full_replay_state)
                   {|
                     l :=
                       existT
                         (dec_exist
                            (sub_index_prop
                               equivocating) i Hi) li;
                     input := input;
                     destination := destination;
                     output := output
                   |}))) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors) simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
match
  match
    equivocator_vlsm_transition_item_project (IM i)
      (composite_transition_item_projection
         (Common.equivocator_IM IM)
         (pre_VLSM_full_projection_trace_item_project
            (composite_type sub_equivocator_IM)
            (composite_type equivocator_IM)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state)
            {|
              l :=
                existT
                  (dec_exist
                     (sub_index_prop equivocating) i
                     Hi) li;
              input := input;
              destination := destination;
              output := output
            |})) (eqv_descriptors i)
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (lift_equivocators_sub_state_to
                    full_replay_state destination);
             output := output
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  rewrite Heqv_l.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
match
  match
    equivocator_vlsm_transition_item_project (IM i)
      (composite_transition_item_projection
         (Common.equivocator_IM IM)
         (pre_VLSM_full_projection_trace_item_project
            (composite_type sub_equivocator_IM)
            (composite_type equivocator_IM)
            (lift_equivocators_sub_label_to
               full_replay_state)
            (lift_equivocators_sub_state_to
               full_replay_state)
            {|
              l :=
                existT
                  (dec_exist
                     (sub_index_prop equivocating) i
                     Hi) li;
              input := input;
              destination := destination;
              output := output
            |})) (Existing n)
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (lift_equivocators_sub_state_to
                    full_replay_state destination);
             output := output
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
match
  match
    match
      equivocator_state_project
        (lift_equivocators_sub_state_to
           full_replay_state destination i) n
    with
    | Some sj =>
        match
          eq_rect_r
            (λ n : index,
               vlabel (Common.equivocator_IM IM n))
            (equivocator_state_append_label (IM i)
               (full_replay_state i) li) eq_refl
        with
        | Spawn sn =>
            if
             decide
               (n =
                equivocator_state_last
                  (lift_equivocators_sub_state_to
                     full_replay_state destination i))
            then Some (None, NewMachine sn)
            else Some (None, Existing n)
        | ContinueWith i0 lx =>
            if decide (i0 = n)
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := input;
                    destination := sj;
                    output := output
                  |}, Existing i0)
            else Some (None, Existing n)
        | ForkWith i0 lx =>
            if
             decide
               (n =
                equivocator_state_last
                  (lift_equivocators_sub_state_to
                     full_replay_state destination i))
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := input;
                    destination := sj;
                    output := output
                  |}, Existing i0)
            else Some (None, Existing n)
        end
    | None => None
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (lift_equivocators_sub_state_to
                    full_replay_state destination);
             output := output
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  destruct_equivocator_state_project
    (lift_equivocators_sub_state_to full_replay_state destination i) n
    lift_n Hlt_n; [|lia].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
lift_n: vstate (IM i)
Hlt_n: n <
equivocator_state_n
  (lift_equivocators_sub_state_to
     full_replay_state destination i)
match
  match
    match
      eq_rect_r
        (λ n : index,
           vlabel (Common.equivocator_IM IM n))
        (equivocator_state_append_label (IM i)
           (full_replay_state i) li) eq_refl
    with
    | Spawn sn =>
        if
         decide
           (n =
            equivocator_state_last
              (lift_equivocators_sub_state_to
                 full_replay_state destination i))
        then Some (None, NewMachine sn)
        else Some (None, Existing n)
    | ContinueWith i0 lx =>
        if decide (i0 = n)
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := lift_n;
                output := output
              |}, Existing i0)
        else Some (None, Existing n)
    | ForkWith i0 lx =>
        if
         decide
           (n =
            equivocator_state_last
              (lift_equivocators_sub_state_to
                 full_replay_state destination i))
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := lift_n;
                output := output
              |}, Existing i0)
        else Some (None, Existing n)
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (lift_equivocators_sub_state_to
                    full_replay_state destination);
             output := output
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  rewrite (lift_equivocators_sub_state_to_sub) with (Hi := Hi).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
lift_n: vstate (IM i)
Hlt_n: n <
equivocator_state_n
  (lift_equivocators_sub_state_to
     full_replay_state destination i)
match
  match
    match
      eq_rect_r
        (λ n : index,
           vlabel (Common.equivocator_IM IM n))
        (equivocator_state_append_label (IM i)
           (full_replay_state i) li) eq_refl
    with
    | Spawn sn =>
        if
         decide
           (n =
            equivocator_state_last
              (equivocator_state_append
                 (full_replay_state i)
                 (destination
                    (dec_exist
                       (sub_index_prop equivocating) i
                       Hi))))
        then Some (None, NewMachine sn)
        else Some (None, Existing n)
    | ContinueWith i0 lx =>
        if decide (i0 = n)
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := lift_n;
                output := output
              |}, Existing i0)
        else Some (None, Existing n)
    | ForkWith i0 lx =>
        if
         decide
           (n =
            equivocator_state_last
              (equivocator_state_append
                 (full_replay_state i)
                 (destination
                    (dec_exist
                       (sub_index_prop equivocating) i
                       Hi))))
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := lift_n;
                output := output
              |}, Existing i0)
        else Some (None, Existing n)
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (lift_equivocators_sub_state_to
                    full_replay_state destination);
             output := output
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  rewrite equivocator_state_append_lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: vstate
  (free_composite_vlsm
     (Common.equivocator_IM IM))
is: composite_state sub_equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
input: option message
destination: state
output: option message
tr: list transition_item
eqv_descriptors: equivocator_descriptors
n: nat
Heqv_l: eqv_descriptors i = Existing n
s_l_n: vstate (IM i)
Hs_l_n: equivocator_state_project
  (full_replay_state i) n = 
Some s_l_n
Hltn: n < equivocator_state_n (full_replay_state i)
Hltsize: equivocator_state_n (full_replay_state i)
≤ equivocator_state_n
    (lift_equivocators_sub_state_to
       full_replay_state destination i)
lift_n: vstate (IM i)
Hlt_n: n <
equivocator_state_n
  (lift_equivocators_sub_state_to
     full_replay_state destination i)
match
  match
    match
      eq_rect_r
        (λ n : index,
           vlabel (Common.equivocator_IM IM n))
        (equivocator_state_append_label (IM i)
           (full_replay_state i) li) eq_refl
    with
    | Spawn sn =>
        if
         decide
           (n =
            equivocator_state_last
              (destination
                 (dec_exist
                    (sub_index_prop equivocating) i Hi)) +
            equivocator_state_n (full_replay_state i))
        then Some (None, NewMachine sn)
        else Some (None, Existing n)
    | ContinueWith i0 lx =>
        if decide (i0 = n)
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := lift_n;
                output := output
              |}, Existing i0)
        else Some (None, Existing n)
    | ForkWith i0 lx =>
        if
         decide
           (n =
            equivocator_state_last
              (destination
                 (dec_exist
                    (sub_index_prop equivocating) i Hi)) +
            equivocator_state_n (full_replay_state i))
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := lift_n;
                output := output
              |}, Existing i0)
        else Some (None, Existing n)
    end
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               Common.equivocators_state_project IM
                 eqv_descriptors
                 (lift_equivocators_sub_state_to
                    full_replay_state destination);
             output := output
           |},
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | Some (None, deqv') =>
      Some
        (None,
        Common.equivocator_descriptors_update IM
          eqv_descriptors i deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], eqv_descriptors)
  destruct li as [sn_d| id li| id li]; simpl
  ; rewrite !decide_False by lia
  ; f_equal; f_equal; apply equivocator_descriptors_update_id; assumption.
Qed.

Lemma equivocators_total_trace_project_replayed_trace_from full_replay_state is tr
  : equivocators_total_trace_project IM (replayed_trace_from full_replay_state is tr) = [].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
equivocators_total_trace_project IM
  (replayed_trace_from full_replay_state is tr) = []
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
equivocators_total_trace_project IM
  (replayed_trace_from full_replay_state is tr) = []
  unfold equivocators_total_trace_project.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
from_option fst []
  (Projections.equivocators_trace_project IM
     (zero_descriptor IM)
     (replayed_trace_from full_replay_state is tr)) =
[]
  rewrite equivocators_trace_project_replayed_trace_from_left
  ; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: state
is: composite_state sub_equivocator_IM
tr: list transition_item
not_equivocating_equivocator_descriptors IM
  (zero_descriptor IM) full_replay_state
  apply zero_descriptor_not_equivocating.
Qed.

Lemma lift_equivocators_sub_valid
  (full_replay_state : composite_state equivocator_IM)
  l s om
  (Hv : composite_valid sub_equivocator_IM l (s, om))
  : composite_valid equivocator_IM (lift_equivocators_sub_label_to full_replay_state  l)
      (lift_equivocators_sub_state_to full_replay_state s, om).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
l: composite_label sub_equivocator_IM
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM l (s, om)
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
l: composite_label sub_equivocator_IM
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM l (s, om)
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  destruct l as (sub_i, li).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
sub_i: sub_index equivocating
li: vlabel (sub_equivocator_IM sub_i)
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM
  (existT sub_i li) 
  (s, om)
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT sub_i li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  destruct_dec_sig sub_i i Hi Heqsub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
sub_i: sub_index equivocating
li: vlabel (sub_equivocator_IM sub_i)
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM
  (existT sub_i li) 
  (s, om)
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT sub_i li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) subst sub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  specialize
    (equivocator_state_append_valid (IM i) li
      (s (dec_exist (sub_index_prop equivocating) i Hi)) om
      (full_replay_state i) Hv
    ) as Hlift.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_valid 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om)
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  cbn.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_valid 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om)
equivocator_valid (IM i)
  (equivocator_state_append_label (IM i)
     (full_replay_state i) li)
  (lift_equivocators_sub_state_to full_replay_state s
     i, om)
  rewrite (lift_equivocators_sub_state_to_sub _ _ _ Hi).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_valid 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om)
equivocator_valid (IM i)
  (equivocator_state_append_label (IM i)
     (full_replay_state i) li)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)),
  om)
  assumption.
Qed.

Lemma lift_equivocators_sub_transition
  (full_replay_state : composite_state equivocator_IM)
  l s om s' om'
  (Hv : composite_valid sub_equivocator_IM l (s, om))
  (Ht: composite_transition sub_equivocator_IM l (s, om) = (s', om'))
  : composite_transition equivocator_IM (lift_equivocators_sub_label_to full_replay_state  l)
      (lift_equivocators_sub_state_to full_replay_state s, om) =
      (lift_equivocators_sub_state_to full_replay_state s', om').message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
l: composite_label sub_equivocator_IM
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Hv: composite_valid sub_equivocator_IM l (s, om)
Ht: composite_transition sub_equivocator_IM l (s, om) =
(s', om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
l: composite_label sub_equivocator_IM
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Hv: composite_valid sub_equivocator_IM l (s, om)
Ht: composite_transition sub_equivocator_IM l (s, om) =
(s', om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
  destruct l as (sub_i, li).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
sub_i: sub_index equivocating
li: vlabel (sub_equivocator_IM sub_i)
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Hv: composite_valid sub_equivocator_IM
  (existT sub_i li) 
  (s, om)
Ht: composite_transition sub_equivocator_IM
  (existT sub_i li) 
  (s, om) = 
(s', om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT sub_i li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
  destruct_dec_sig sub_i i Hi Heqsub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
sub_i: sub_index equivocating
li: vlabel (sub_equivocator_IM sub_i)
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Hv: composite_valid sub_equivocator_IM
  (existT sub_i li) 
  (s, om)
Ht: composite_transition sub_equivocator_IM
  (existT sub_i li) 
  (s, om) = 
(s', om')
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT sub_i li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om') subst sub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Ht: composite_transition sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om) = 
(s', om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
  specialize
    (equivocator_state_append_transition (IM i) li
      (s (dec_exist (sub_index_prop equivocating) i Hi)) om
      (s' (dec_exist (sub_index_prop equivocating) i Hi)) om'
      (full_replay_state i) Hv
    ) as Hlift.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Ht: composite_transition sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om) = 
(s', om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i) li
  (s
     (dec_exist 
        (sub_index_prop equivocating) i Hi),
  om) =
(s'
   (dec_exist 
      (sub_index_prop equivocating) i Hi),
om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i)
     (s'
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
  cbn in Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li
     (s
        (dec_exist 
         (sub_index_prop equivocating) i Hi), om) in
 (state_update sub_equivocator_IM s
    (dec_exist (sub_index_prop equivocating) i Hi)
    si', om')) = 
(s', om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i) li
  (s
     (dec_exist 
        (sub_index_prop equivocating) i Hi),
  om) =
(s'
   (dec_exist 
      (sub_index_prop equivocating) i Hi),
om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i)
     (s'
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
  destruct (equivocator_transition _ _ _) as (_si', _om').message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
_si': equivocator_state (IM i)
_om': option message
Ht: (state_update sub_equivocator_IM s
   (dec_exist (sub_index_prop equivocating) i Hi)
   _si', _om') = 
(s', om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: (_si', _om') =
(s'
   (dec_exist 
      (sub_index_prop equivocating) i Hi),
om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i)
     (s'
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
  inversion Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
s': composite_state sub_equivocator_IM
om': option message
_si': equivocator_state (IM i)
_om': option message
Ht: (state_update sub_equivocator_IM s
   (dec_exist (sub_index_prop equivocating) i Hi)
   _si', _om') = 
(s', om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: (_si', _om') =
(s'
   (dec_exist 
      (sub_index_prop equivocating) i Hi),
om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i)
     (s'
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om')
H0: state_update sub_equivocator_IM s
  (dec_exist (sub_index_prop equivocating) i Hi)
  _si' = s'
H1: _om' = om'
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), om') subst s' om'.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Ht: (state_update sub_equivocator_IM s
   (dec_exist (sub_index_prop equivocating) i Hi)
   _si', _om') =
(state_update sub_equivocator_IM s
   (dec_exist (sub_index_prop equivocating) i Hi)
   _si', _om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: (_si', _om') =
(state_update sub_equivocator_IM s
   (dec_exist 
      (sub_index_prop equivocating) i Hi) _si'
   (dec_exist 
      (sub_index_prop equivocating) i Hi),
_om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i)
     (state_update sub_equivocator_IM s
        (dec_exist
         (sub_index_prop equivocating) i Hi)
        _si'
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  _om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om') clear Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: (_si', _om') =
(state_update sub_equivocator_IM s
   (dec_exist 
      (sub_index_prop equivocating) i Hi) _si'
   (dec_exist 
      (sub_index_prop equivocating) i Hi),
_om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i)
     (state_update sub_equivocator_IM s
        (dec_exist
         (sub_index_prop equivocating) i Hi)
        _si'
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  _om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om') rewrite state_update_eq in Hlift.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: (_si', _om') = (_si', _om')
→ equivocator_transition 
    (IM i)
    (equivocator_state_append_label 
       (IM i) 
       (full_replay_state i) li)
    (equivocator_state_append
       (full_replay_state i)
       (s
         (dec_exist
         (sub_index_prop equivocating) i Hi)),
    om) =
  (equivocator_state_append
     (full_replay_state i) _si', _om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om')
  specialize (Hlift eq_refl).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
composite_transition equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state
     (existT
        (dec_exist (sub_index_prop equivocating) i Hi)
        li))
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om')
  cbn.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
(let (si', om') :=
   equivocator_transition (IM i)
     (equivocator_state_append_label (IM i)
        (full_replay_state i) li)
     (lift_equivocators_sub_state_to full_replay_state
        s i, om) in
 (state_update equivocator_IM
    (lift_equivocators_sub_state_to full_replay_state
       s) i si', om')) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om')
  rewrite (lift_equivocators_sub_state_to_sub _ _ _ Hi).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
(let (si', om') :=
   equivocator_transition (IM i)
     (equivocator_state_append_label (IM i)
        (full_replay_state i) li)
     (equivocator_state_append (full_replay_state i)
        (s
           (dec_exist (sub_index_prop equivocating) i
              Hi)), om) in
 (state_update equivocator_IM
    (lift_equivocators_sub_state_to full_replay_state
       s) i si', om')) =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om')
  replace (equivocator_transition _ _ _) with
    (equivocator_state_append (full_replay_state i) _si', _om').message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
(state_update equivocator_IM
   (lift_equivocators_sub_state_to full_replay_state s)
   i
   (equivocator_state_append (full_replay_state i)
      _si'), _om') =
(lift_equivocators_sub_state_to full_replay_state
   (state_update sub_equivocator_IM s
      (dec_exist (sub_index_prop equivocating) i Hi)
      _si'), _om')
  f_equal.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  i
  (equivocator_state_append (full_replay_state i) _si') =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si')
  apply functional_extensionality_dep.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
∀ x : index,
  state_update equivocator_IM
    (lift_equivocators_sub_state_to full_replay_state
       s) i
    (equivocator_state_append (full_replay_state i)
       _si') x =
  lift_equivocators_sub_state_to full_replay_state
    (state_update sub_equivocator_IM s
       (dec_exist (sub_index_prop equivocating) i Hi)
       _si') x intro j.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  i
  (equivocator_state_append (full_replay_state i) _si')
  j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si') j
  destruct (decide (i = j)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
e: i = j
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  i
  (equivocator_state_append (full_replay_state i) _si')
  j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si') j
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  i
  (equivocator_state_append (full_replay_state i) _si')
  j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si') j
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
e: i = j
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  i
  (equivocator_state_append (full_replay_state i) _si')
  j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si') j subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
j: index
Hi: sub_index_prop equivocating j
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) j Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM j)
_om': option message
Hlift: equivocator_transition 
  (IM j)
  (equivocator_state_append_label 
     (IM j) 
     (full_replay_state j) li)
  (equivocator_state_append
     (full_replay_state j)
     (s
        (dec_exist
         (sub_index_prop equivocating) j Hi)),
  om) =
(equivocator_state_append
   (full_replay_state j) _si', _om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) j Hi) li)
  (s, om)
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  j
  (equivocator_state_append (full_replay_state j) _si')
  j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) j Hi)
     _si') j rewrite state_update_eq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
j: index
Hi: sub_index_prop equivocating j
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) j Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM j)
_om': option message
Hlift: equivocator_transition 
  (IM j)
  (equivocator_state_append_label 
     (IM j) 
     (full_replay_state j) li)
  (equivocator_state_append
     (full_replay_state j)
     (s
        (dec_exist
         (sub_index_prop equivocating) j Hi)),
  om) =
(equivocator_state_append
   (full_replay_state j) _si', _om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) j Hi) li)
  (s, om)
equivocator_state_append (full_replay_state j) _si' =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) j Hi)
     _si') j
    rewrite (lift_equivocators_sub_state_to_sub _ _ _ Hi).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
j: index
Hi: sub_index_prop equivocating j
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) j Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM j)
_om': option message
Hlift: equivocator_transition 
  (IM j)
  (equivocator_state_append_label 
     (IM j) 
     (full_replay_state j) li)
  (equivocator_state_append
     (full_replay_state j)
     (s
        (dec_exist
         (sub_index_prop equivocating) j Hi)),
  om) =
(equivocator_state_append
   (full_replay_state j) _si', _om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) j Hi) li)
  (s, om)
equivocator_state_append (full_replay_state j) _si' =
equivocator_state_append (full_replay_state j)
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) j Hi)
     _si'
     (dec_exist (sub_index_prop equivocating) j Hi))
    rewrite state_update_eq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
j: index
Hi: sub_index_prop equivocating j
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) j Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM j)
_om': option message
Hlift: equivocator_transition 
  (IM j)
  (equivocator_state_append_label 
     (IM j) 
     (full_replay_state j) li)
  (equivocator_state_append
     (full_replay_state j)
     (s
        (dec_exist
         (sub_index_prop equivocating) j Hi)),
  om) =
(equivocator_state_append
   (full_replay_state j) _si', _om')
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) j Hi) li)
  (s, om)
equivocator_state_append (full_replay_state j) _si' =
equivocator_state_append (full_replay_state j) _si' reflexivity.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
state_update equivocator_IM
  (lift_equivocators_sub_state_to full_replay_state s)
  i
  (equivocator_state_append (full_replay_state i) _si')
  j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si') j rewrite state_update_neq by congruence.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
lift_equivocators_sub_state_to full_replay_state s j =
lift_equivocators_sub_state_to full_replay_state
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si') j
    unfold lift_equivocators_sub_state_to.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
match decide (sub_index_prop equivocating j) with
| left e =>
    equivocator_state_append (full_replay_state j)
      (s (dec_exist (sub_index_prop equivocating) j e))
| in_right => full_replay_state j
end =
match decide (sub_index_prop equivocating j) with
| left e =>
    equivocator_state_append (full_replay_state j)
      (state_update sub_equivocator_IM s
         (dec_exist (sub_index_prop equivocating) i Hi)
         _si'
         (dec_exist (sub_index_prop equivocating) j e))
| in_right => full_replay_state j
end
    destruct (decide _); [|reflexivity].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
s0: sub_index_prop equivocating j
equivocator_state_append (full_replay_state j)
  (s (dec_exist (sub_index_prop equivocating) j s0)) =
equivocator_state_append (full_replay_state j)
  (state_update sub_equivocator_IM s
     (dec_exist (sub_index_prop equivocating) i Hi)
     _si'
     (dec_exist (sub_index_prop equivocating) j s0))
    rewrite state_update_neq; [reflexivity|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
s0: sub_index_prop equivocating j
dec_exist (sub_index_prop equivocating) j s0
≠ dec_exist (sub_index_prop equivocating) i Hi
    intro Hcontra.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
s0: sub_index_prop equivocating j
Hcontra: dec_exist (sub_index_prop equivocating) j s0 =
dec_exist (sub_index_prop equivocating) i Hi
False apply dsig_eq in Hcontra.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
s0: sub_index_prop equivocating j
Hcontra: `
(dec_exist 
   (sub_index_prop equivocating) j s0) =
`
(dec_exist 
   (sub_index_prop equivocating) i Hi)
False simpl in Hcontra.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
i: index
Hi: sub_index_prop equivocating i
li: vlabel
  (sub_equivocator_IM
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
s: composite_state sub_equivocator_IM
om: option message
_si': equivocator_state (IM i)
_om': option message
Hv: composite_valid sub_equivocator_IM
  (existT
     (dec_exist 
        (sub_index_prop equivocating) i Hi) li)
  (s, om)
Hlift: equivocator_transition 
  (IM i)
  (equivocator_state_append_label 
     (IM i) 
     (full_replay_state i) li)
  (equivocator_state_append
     (full_replay_state i)
     (s
        (dec_exist
         (sub_index_prop equivocating) i Hi)),
  om) =
(equivocator_state_append
   (full_replay_state i) _si', _om')
j: index
n: i ≠ j
s0: sub_index_prop equivocating j
Hcontra: j = i
False congruence.
Qed.

Section pre_loaded_constrained_projection.

By replaying a valid_trace on top of a valid_state we obtain a valid_trace. We derive this as a more general VLSM_weak_full_projection result for a class of VLSM parameterized by a constraint having "good" properties and pre-loaded with a seed, to allow deriving the VLSM_weak_full_projection result for both the free composition of equivocators and for the no message equivocation composition of equivocators (free, or with an additional fixed-set state-equivocation constraint).

Context
  (constraint : composite_label equivocator_IM -> composite_state equivocator_IM * option message -> Prop)
  (seed1 : message -> Prop)
  (SeededCE := pre_loaded_vlsm (composite_vlsm equivocator_IM constraint) seed1)
  (Hconstraint_none : forall i ns s, i ∈ equivocating -> valid_state_prop SeededCE s -> constraint (existT i (Spawn ns)) (s, None))
  (Hseed : forall m, seed m -> valid_message_prop SeededCE m)
  (full_replay_state : composite_state equivocator_IM)
  (Hfull_replay_state : valid_state_prop SeededCE full_replay_state)
  (Hsubsumption : forall l s om, input_valid SeededXE l (s, om) ->
    valid_state_prop SeededCE (lift_equivocators_sub_state_to full_replay_state s) ->
    constraint (lift_equivocators_sub_label_to full_replay_state l)  (lift_equivocators_sub_state_to full_replay_state s, om))
  .

Lemma replayed_initial_state_from_valid
  (is : composite_state sub_equivocator_IM)
  (His : composite_initial_state_prop sub_equivocator_IM is)
  : finite_valid_trace_from SeededCE full_replay_state (replayed_initial_state_from full_replay_state is).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state is)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state is)
  cut (forall l, incl l (@sub_index_listing _ _ equivocating index_listing) ->
    finite_valid_plan_from SeededCE
      full_replay_state (map (initial_new_machine_transition_item is) l)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
(∀ l : list (sub_index equivocating),
   incl l (sub_index_listing equivocating)
   → finite_valid_plan_from SeededCE full_replay_state
       (map (initial_new_machine_transition_item is) l))
→ finite_valid_trace_from SeededCE full_replay_state
    (replayed_initial_state_from full_replay_state is)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
∀ l : list (sub_index equivocating),
  incl l (sub_index_listing equivocating)
  → finite_valid_plan_from SeededCE full_replay_state
      (map (initial_new_machine_transition_item is) l)
  {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
(∀ l : list (sub_index equivocating),
   incl l (sub_index_listing equivocating)
   → finite_valid_plan_from SeededCE full_replay_state
       (map (initial_new_machine_transition_item is) l))
→ finite_valid_trace_from SeededCE full_replay_state
    (replayed_initial_state_from full_replay_state is) intros Hplan.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hplan: ∀ l : list (sub_index equivocating),
  incl l (sub_index_listing equivocating)
  → finite_valid_plan_from SeededCE
      full_replay_state
      (map
         (initial_new_machine_transition_item
         is) l)
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state is) specialize (Hplan _ (incl_refl _)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hplan: finite_valid_plan_from SeededCE
  full_replay_state
  (map
     (initial_new_machine_transition_item is)
     (sub_index_listing equivocating))
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state is)
    unfold finite_valid_plan_from in Hplan.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
Hplan: finite_valid_trace_from SeededCE
  full_replay_state
  (apply_plan SeededCE full_replay_state
     (map
        (initial_new_machine_transition_item
         is) 
        (sub_index_listing equivocating))).1
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state is)
    assumption.
  }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
∀ l : list (sub_index equivocating),
  incl l (sub_index_listing equivocating)
  → finite_valid_plan_from SeededCE full_replay_state
      (map (initial_new_machine_transition_item is) l)
  intro l.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
l: list (sub_index equivocating)
incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE full_replay_state
    (map (initial_new_machine_transition_item is) l)
  induction l using rev_ind; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: incl [] (sub_index_listing equivocating)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is) [])
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is)
     (l ++ [x]))
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: incl [] (sub_index_listing equivocating)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is) []) constructor.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
H: incl [] (sub_index_listing equivocating)
valid_state_prop SeededCE full_replay_state assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is)
     (l ++ [x])) spec IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
incl l (sub_index_listing equivocating)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is)
     (l ++ [x]))  {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
incl l (sub_index_listing equivocating)  intros i Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
i: sub_index equivocating
Hi: In i l
In i (sub_index_listing equivocating) apply H.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
i: sub_index equivocating
Hi: In i l
In i (l ++ [x]) apply in_app_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
i: sub_index equivocating
Hi: In i l
In i l ∨ In i [x] left.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
IHl: incl l (sub_index_listing equivocating)
→ finite_valid_plan_from SeededCE
    full_replay_state
    (map
       (initial_new_machine_transition_item is)
       l)
H: incl (l ++ [x]) (sub_index_listing equivocating)
i: sub_index equivocating
Hi: In i l
In i l assumption. }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is)
     (l ++ [x]))
    rewrite map_app.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is) l ++
   map (initial_new_machine_transition_item is) [x])
    apply finite_valid_plan_from_app_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is) l)
∧ finite_valid_plan_from SeededCE
    (apply_plan SeededCE full_replay_state
       (map (initial_new_machine_transition_item is) l)).2
    (map (initial_new_machine_transition_item is) [x])
    split; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
finite_valid_plan_from SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2
  (map (initial_new_machine_transition_item is) [x])
    apply finite_valid_trace_singleton.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
input_valid_transition SeededCE
  (existT (` x)
     (Spawn (equivocator_state_zero (is x))))
  ((apply_plan SeededCE full_replay_state
      (map (initial_new_machine_transition_item is) l)).2,
  None)
  (state_update equivocator_IM
     (apply_plan SeededCE full_replay_state
        (map (initial_new_machine_transition_item is)
           l)).2 (` x)
     (equivocator_state_extend
        ((apply_plan SeededCE full_replay_state
            (map
               (initial_new_machine_transition_item is)
               l)).2 (` x), None).1
        (equivocator_state_zero (is x))), None) simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
input_valid_transition SeededCE
  (existT (` x)
     (Spawn (equivocator_state_zero (is x))))
  ((apply_plan SeededCE full_replay_state
      (map (initial_new_machine_transition_item is) l)).2,
  None)
  (state_update equivocator_IM
     (apply_plan SeededCE full_replay_state
        (map (initial_new_machine_transition_item is)
           l)).2 (` x)
     (equivocator_state_extend
        ((apply_plan SeededCE full_replay_state
            (map
               (initial_new_machine_transition_item is)
               l)).2 (` x))
        (equivocator_state_zero (is x))), None)
    repeat split.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
valid_state_prop SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
option_valid_message_prop SeededCE None
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
vinitial_state_prop 
  (IM (` x)) 
  (equivocator_state_zero (is x))
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
constraint
  (existT 
     (` x) 
     (Spawn (equivocator_state_zero (is x))))
  ((apply_plan SeededCE full_replay_state
      (map (initial_new_machine_transition_item is) l)).2,
  None)
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
valid_state_prop SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2 revert IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
finite_valid_plan_from SeededCE full_replay_state
  (map (initial_new_machine_transition_item is) l)
→ valid_state_prop SeededCE
    (apply_plan SeededCE full_replay_state
       (map (initial_new_machine_transition_item is) l)).2 apply apply_plan_last_valid.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
option_valid_message_prop SeededCE None apply option_valid_message_None.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
vinitial_state_prop (IM (` x))
  (equivocator_state_zero (is x)) apply His.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
constraint
  (existT (` x)
     (Spawn (equivocator_state_zero (is x))))
  ((apply_plan SeededCE full_replay_state
      (map (initial_new_machine_transition_item is) l)).2,
  None) apply Hconstraint_none.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
` x ∈ equivocating
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
valid_state_prop SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2
      *message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
` x ∈ equivocating clear -x.index: Type
IndEqDec: EqDecision index
equivocating: list index
x: sub_index equivocating
` x ∈ equivocating destruct_dec_sig x i Hi Hx.index: Type
IndEqDec: EqDecision index
equivocating: list index
x: sub_index equivocating
i: index
Hi: sub_index_prop equivocating i
Hx: x = dec_exist (sub_index_prop equivocating) i Hi
` x ∈ equivocating
        subst.index: Type
IndEqDec: EqDecision index
equivocating: list index
i: index
Hi: sub_index_prop equivocating i
` (dec_exist (sub_index_prop equivocating) i Hi)
∈ equivocating assumption.
      *message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: finite_valid_plan_from SeededCE
  full_replay_state
  (map (initial_new_machine_transition_item is)
     l)
valid_state_prop SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2 apply finite_valid_trace_last_pstate in IHl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: valid_state_prop SeededCE
  (finite_trace_last full_replay_state
     (apply_plan SeededCE full_replay_state
        (map
         (initial_new_machine_transition_item is)
         l)).1)
valid_state_prop SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2
        remember (finite_trace_last _ _) as lst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
lst: state
Heqlst: lst =
finite_trace_last full_replay_state
  (apply_plan SeededCE full_replay_state
     (map
        (initial_new_machine_transition_item
         is) l)).1
IHl: valid_state_prop SeededCE lst
valid_state_prop SeededCE
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).2
        replace ((apply_plan _ _ _).2) with lst
        ; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
lst: state
Heqlst: lst =
finite_trace_last full_replay_state
  (apply_plan SeededCE full_replay_state
     (map
        (initial_new_machine_transition_item
         is) l)).1
IHl: valid_state_prop SeededCE lst
lst =
(apply_plan SeededCE full_replay_state
   (map (initial_new_machine_transition_item is) l)).2
        subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
His: composite_initial_state_prop sub_equivocator_IM
  is
x: sub_index equivocating
l: list (sub_index equivocating)
H: incl (l ++ [x]) (sub_index_listing equivocating)
IHl: valid_state_prop SeededCE
  (finite_trace_last full_replay_state
     (apply_plan SeededCE full_replay_state
        (map
         (initial_new_machine_transition_item is)
         l)).1)
finite_trace_last full_replay_state
  (apply_plan SeededCE full_replay_state
     (map (initial_new_machine_transition_item is) l)).1 =
(apply_plan SeededCE full_replay_state
   (map (initial_new_machine_transition_item is) l)).2
        apply apply_plan_last.
Qed.

Lemma lift_initial_message
  : forall m, vinitial_message_prop SeededXE m -> valid_message_prop SeededCE m.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
∀ m : message,
  vinitial_message_prop SeededXE m
  → valid_message_prop SeededCE m
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
∀ m : message,
  vinitial_message_prop SeededXE m
  → valid_message_prop SeededCE m
  intros m [Hinit | Hseeded].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
Hinit: initial_message_prop m
valid_message_prop SeededCE m
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
Hseeded: seed m
valid_message_prop SeededCE m
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
Hinit: initial_message_prop m
valid_message_prop SeededCE m apply initial_message_is_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
Hinit: initial_message_prop m
initial_message_prop m destruct Hinit as [[i Hi] Hinit].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
i: index
Hi: bool_decide (sub_index_prop equivocating i)
Hinit: ∃ mi : vinitial_message
         (SubProjectionTraces.sub_IM
         (Common.equivocator_IM IM)
         equivocating 
         (i ↾ Hi)), 
  ` mi = m
initial_message_prop m
    left.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
i: index
Hi: bool_decide (sub_index_prop equivocating i)
Hinit: ∃ mi : vinitial_message
         (SubProjectionTraces.sub_IM
         (Common.equivocator_IM IM)
         equivocating 
         (i ↾ Hi)), 
  ` mi = m
initial_message_prop m exists i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
i: index
Hi: bool_decide (sub_index_prop equivocating i)
Hinit: ∃ mi : vinitial_message
         (SubProjectionTraces.sub_IM
         (Common.equivocator_IM IM)
         equivocating 
         (i ↾ Hi)), 
  ` mi = m
∃ mi : vinitial_message (equivocator_IM i), ` mi = m assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
Hseeded: seed m
valid_message_prop SeededCE m apply Hseed.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
m: message
Hseeded: seed m
seed m assumption.
Qed.

Lemma lift_equivocators_sub_weak_projection
  : VLSM_weak_full_projection SeededXE SeededCE (lift_equivocators_sub_label_to full_replay_state) (lift_equivocators_sub_state_to full_replay_state).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
VLSM_weak_full_projection SeededXE SeededCE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
VLSM_weak_full_projection SeededXE SeededCE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
  apply basic_VLSM_weak_full_projection; intro; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
vvalid SeededCE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition SeededXE l 
  (s, om) 
  (s', om')
vtransition SeededCE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
s: state
H: vinitial_state_prop SeededXE s
valid_state_prop SeededCE
  (lift_equivocators_sub_state_to full_replay_state s)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HmX: vinitial_message_prop SeededXE m
valid_message_prop SeededCE m
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
vvalid SeededCE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) split.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
constraint
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
composite_valid equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) apply lift_equivocators_sub_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
composite_valid sub_equivocator_IM l (s, om) apply Hv.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
Hv: input_valid SeededXE l (s, om)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop SeededCE om
constraint
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) apply Hsubsumption; assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition SeededXE l 
  (s, om) 
  (s', om')
vtransition SeededCE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om') apply lift_equivocators_sub_transition; apply H.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
s: state
H: vinitial_state_prop SeededXE s
valid_state_prop SeededCE
  (lift_equivocators_sub_state_to full_replay_state s) rewrite <- replayed_initial_state_from_lift; [|assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
s: state
H: vinitial_state_prop SeededXE s
valid_state_prop SeededCE
  (finite_trace_last full_replay_state
     (replayed_initial_state_from full_replay_state s))
    apply finite_valid_trace_last_pstate.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
s: state
H: vinitial_state_prop SeededXE s
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state s)
    by apply replayed_initial_state_from_valid.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
l: label
s: state
m: message
Hv: input_valid SeededXE l (s, Some m)
HsY: valid_state_prop SeededCE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HmX: vinitial_message_prop SeededXE m
valid_message_prop SeededCE m by apply lift_initial_message.
Qed.

Lemma sub_preloaded_replayed_trace_from_valid_equivocating
  (is : composite_state sub_equivocator_IM)
  (tr : list (composite_transition_item sub_equivocator_IM))
  (Htr : finite_valid_trace SeededXE is tr)
  : finite_valid_trace_from SeededCE
      full_replay_state (replayed_trace_from full_replay_state is tr).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededCE full_replay_state
  (replayed_trace_from full_replay_state is tr)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededCE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  destruct Htr as [Htr His].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace_from SeededXE is tr
His: initial_state_prop is
finite_valid_trace_from SeededCE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  apply finite_valid_trace_from_app_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace_from SeededXE is tr
His: initial_state_prop is
finite_valid_trace_from SeededCE full_replay_state
  (replayed_initial_state_from full_replay_state is)
∧ finite_valid_trace_from SeededCE
    (finite_trace_last full_replay_state
       (replayed_initial_state_from full_replay_state
          is))
    (pre_VLSM_full_projection_finite_trace_project
       (type FreeSubE) (type FreeE)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state) tr)
  split; [apply replayed_initial_state_from_valid; assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace_from SeededXE is tr
His: initial_state_prop is
finite_valid_trace_from SeededCE
  (finite_trace_last full_replay_state
     (replayed_initial_state_from full_replay_state is))
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) (type FreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     tr)
  rewrite replayed_initial_state_from_lift by assumption.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace_from SeededXE is tr
His: initial_state_prop is
finite_valid_trace_from SeededCE
  (lift_equivocators_sub_state_to full_replay_state is)
  (pre_VLSM_full_projection_finite_trace_project
     (type FreeSubE) (type FreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     tr)
  revert Htr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
constraint: composite_label equivocator_IM
→ composite_state equivocator_IM *
  option message → Prop
seed1: message → Prop
SeededCE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM constraint)
  seed1: VLSM message
Hconstraint_none: ∀ (i : index) 
  (ns : vstate (IM i)) 
  (s : state),
  i ∈ equivocating
  → valid_state_prop SeededCE s
    → constraint
        (existT i (Spawn ns))
        (s, None)
Hseed: ∀ m : message,
  seed m → valid_message_prop SeededCE m
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededCE
  full_replay_state
Hsubsumption: ∀ (l : label) 
  (s : state) 
  (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop SeededCE
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → constraint
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
His: initial_state_prop is
finite_valid_trace_from SeededXE is tr
→ finite_valid_trace_from SeededCE
    (lift_equivocators_sub_state_to full_replay_state
       is)
    (pre_VLSM_full_projection_finite_trace_project
       (type FreeSubE) (type FreeE)
       (lift_equivocators_sub_label_to
          full_replay_state)
       (lift_equivocators_sub_state_to
          full_replay_state) tr)
  apply (VLSM_weak_full_projection_finite_valid_trace_from lift_equivocators_sub_weak_projection).
Qed.

End pre_loaded_constrained_projection.

Lemma SeededXE_PreFreeE_weak_full_projection
  (full_replay_state : composite_state equivocator_IM)
  (Hfull_replay_state : valid_state_prop PreFreeE  full_replay_state)
  : VLSM_weak_full_projection SeededXE PreFreeE (lift_equivocators_sub_label_to full_replay_state) (lift_equivocators_sub_state_to full_replay_state).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
VLSM_weak_full_projection SeededXE PreFreeE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
VLSM_weak_full_projection SeededXE PreFreeE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
  constructor.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
∀ (sX : state) (trX : list transition_item),
  finite_valid_trace_from SeededXE sX trX
  → finite_valid_trace_from PreFreeE
      (lift_equivocators_sub_state_to
         full_replay_state sX)
      (pre_VLSM_full_projection_finite_trace_project
         (type SeededXE) (type PreFreeE)
         (lift_equivocators_sub_label_to
            full_replay_state)
         (lift_equivocators_sub_state_to
            full_replay_state) trX)
  intros sX trX HtrX.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
sX: state
trX: list transition_item
HtrX: finite_valid_trace_from SeededXE sX trX
finite_valid_trace_from PreFreeE
  (lift_equivocators_sub_state_to full_replay_state sX)
  (pre_VLSM_full_projection_finite_trace_project
     (type SeededXE) (type PreFreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     trX)
  specialize (pre_loaded_with_all_messages_vlsm_is_pre_loaded_with_True FreeE) as Heq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
sX: state
trX: list transition_item
HtrX: finite_valid_trace_from SeededXE sX trX
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
finite_valid_trace_from PreFreeE
  (lift_equivocators_sub_state_to full_replay_state sX)
  (pre_VLSM_full_projection_finite_trace_project
     (type SeededXE) (type PreFreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     trX)
  apply (VLSM_eq_finite_valid_trace_from Heq).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
sX: state
trX: list transition_item
HtrX: finite_valid_trace_from SeededXE sX trX
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
finite_valid_trace_from
  (mk_vlsm
     (machine
        (pre_loaded_vlsm FreeE (λ _ : message, True))))
  (lift_equivocators_sub_state_to full_replay_state sX)
  (pre_VLSM_full_projection_finite_trace_project
     (type SeededXE) (type PreFreeE)
     (lift_equivocators_sub_label_to full_replay_state)
     (lift_equivocators_sub_state_to full_replay_state)
     trX)
  revert sX trX HtrX.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (sX : state) (trX : list transition_item),
  finite_valid_trace_from SeededXE sX trX
  → finite_valid_trace_from
      (mk_vlsm
         (machine
            (pre_loaded_vlsm FreeE
               (λ _ : message, True))))
      (lift_equivocators_sub_state_to
         full_replay_state sX)
      (pre_VLSM_full_projection_finite_trace_project
         (type SeededXE) (type PreFreeE)
         (lift_equivocators_sub_label_to
            full_replay_state)
         (lift_equivocators_sub_state_to
            full_replay_state) trX)
  apply lift_equivocators_sub_weak_projection.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (i : index) (ns : vstate (IM i)) (s : state),
  i ∈ equivocating
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (free_constraint equivocator_IM))
         (λ _ : message, True)) s
    → free_constraint equivocator_IM
        (existT i (Spawn ns)) (s, None)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ m : message,
  seed m
  → valid_message_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (free_constraint equivocator_IM))
         (λ _ : message, True)) m
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) full_replay_state
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (l : label) (s : state) (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (free_constraint equivocator_IM))
         (λ _ : message, True))
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → free_constraint equivocator_IM
        (lift_equivocators_sub_label_to
         full_replay_state l)
        (lift_equivocators_sub_state_to
         full_replay_state s, om)
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (i : index) (ns : vstate (IM i)) (s : state),
  i ∈ equivocating
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (free_constraint equivocator_IM))
         (λ _ : message, True)) s
    → free_constraint equivocator_IM
        (existT i (Spawn ns)) (s, None) intros; exact I.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ m : message,
  seed m
  → valid_message_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (free_constraint equivocator_IM))
         (λ _ : message, True)) m intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
m: message
H: seed m
valid_message_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) m apply initial_message_is_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
m: message
H: seed m
initial_message_prop m right.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
m: message
H: seed m
True exact I.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) full_replay_state apply (VLSM_eq_valid_state Heq) in Hfull_replay_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_vlsm FreeE
         (λ _ : message, True))))
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) full_replay_state assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (l : label) (s : state) (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (free_constraint equivocator_IM))
         (λ _ : message, True))
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → free_constraint equivocator_IM
        (lift_equivocators_sub_label_to
           full_replay_state l)
        (lift_equivocators_sub_state_to
           full_replay_state s, om) intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
l: label
s: state
om: option message
H: input_valid SeededXE l (s, om)
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True))
  (lift_equivocators_sub_state_to
     full_replay_state s)
free_constraint equivocator_IM
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) exact I.
Qed.

Lemma PreFreeSubE_PreFreeE_weak_full_projection
  (full_replay_state : composite_state equivocator_IM)
  (Hfull_replay_state : valid_state_prop PreFreeE  full_replay_state)
  : VLSM_weak_full_projection PreFreeSubE PreFreeE (lift_equivocators_sub_label_to full_replay_state) (lift_equivocators_sub_state_to full_replay_state).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
VLSM_weak_full_projection PreFreeSubE PreFreeE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
VLSM_weak_full_projection PreFreeSubE PreFreeE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
  apply basic_VLSM_weak_full_projection; intro; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
l: label
s: state
om: option message
Hv: input_valid PreFreeSubE l (s, om)
HsY: valid_state_prop PreFreeE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop PreFreeE om
vvalid PreFreeE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition PreFreeSubE l 
  (s, om) 
  (s', om')
vtransition PreFreeE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om')
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
valid_state_prop PreFreeE
  (lift_equivocators_sub_state_to full_replay_state s)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
l: label
s: state
m: message
Hv: input_valid PreFreeSubE l (s, Some m)
HsY: valid_state_prop PreFreeE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HmX: vinitial_message_prop PreFreeSubE m
valid_message_prop PreFreeE m
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
l: label
s: state
om: option message
Hv: input_valid PreFreeSubE l (s, om)
HsY: valid_state_prop PreFreeE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HomY: option_valid_message_prop PreFreeE om
vvalid PreFreeE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) split; [apply lift_equivocators_sub_valid; apply Hv|exact I].
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition PreFreeSubE l 
  (s, om) 
  (s', om')
vtransition PreFreeE
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) =
(lift_equivocators_sub_state_to full_replay_state s',
om') apply lift_equivocators_sub_transition; apply H.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
valid_state_prop PreFreeE
  (lift_equivocators_sub_state_to full_replay_state s) rewrite <- replayed_initial_state_from_lift; [|assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
valid_state_prop PreFreeE
  (finite_trace_last full_replay_state
     (replayed_initial_state_from full_replay_state s))
    apply finite_valid_trace_last_pstate.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
finite_valid_trace_from PreFreeE full_replay_state
  (replayed_initial_state_from full_replay_state s)
    specialize (pre_loaded_with_all_messages_vlsm_is_pre_loaded_with_True FreeE) as Heq.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
finite_valid_trace_from PreFreeE full_replay_state
  (replayed_initial_state_from full_replay_state s)
    apply (VLSM_eq_finite_valid_trace_from Heq).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
finite_valid_trace_from
  (mk_vlsm
     (machine
        (pre_loaded_vlsm FreeE (λ _ : message, True))))
  full_replay_state
  (replayed_initial_state_from full_replay_state s)
    apply replayed_initial_state_from_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (i : index) (ns : vstate (IM i)) (s : state),
  i ∈ equivocating
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (free_constraint equivocator_IM))
         (λ _ : message, True)) s
    → free_constraint equivocator_IM
        (existT i (Spawn ns)) (s, None)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) full_replay_state
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
composite_initial_state_prop sub_equivocator_IM s
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
∀ (i : index) (ns : vstate (IM i)) (s : state),
  i ∈ equivocating
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (free_constraint equivocator_IM))
         (λ _ : message, True)) s
    → free_constraint equivocator_IM
        (existT i (Spawn ns)) (s, None) intro; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
i: index
ns: vstate (IM i)
s0: state
H0: i ∈ equivocating
H1: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) s0
free_constraint equivocator_IM (existT i (Spawn ns))
  (s0, None) exact I.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (free_constraint equivocator_IM))
     (λ _ : message, True)) full_replay_state apply (VLSM_eq_valid_state Heq).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm FreeE)))
  full_replay_state assumption.
    +message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
s: state
H: vinitial_state_prop PreFreeSubE s
Heq: VLSM_eq
  (pre_loaded_with_all_messages_vlsm FreeE)
  (pre_loaded_vlsm FreeE (λ _ : message, True))
composite_initial_state_prop sub_equivocator_IM s assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop PreFreeE
  full_replay_state
l: label
s: state
m: message
Hv: input_valid PreFreeSubE l (s, Some m)
HsY: valid_state_prop PreFreeE
  (lift_equivocators_sub_state_to
     full_replay_state s)
HmX: vinitial_message_prop PreFreeSubE m
valid_message_prop PreFreeE m apply any_message_is_valid_in_preloaded.
Qed.

Section seeded_no_equiv.

Context
  (SeededAllXE : VLSM message := composite_no_equivocation_vlsm_with_pre_loaded equivocator_IM (free_constraint _) (equivocator_Hbs IM Hbs) seed)
  (full_replay_state : composite_state equivocator_IM)
  (Hfull_replay_state : valid_state_prop SeededAllXE full_replay_state)
  .

Local Lemma SeededNoEquiv_subsumption
  : forall l s om, input_valid SeededXE l (s, om) ->
  no_equivocations_additional_constraint_with_pre_loaded equivocator_IM (free_constraint _) (equivocator_Hbs IM Hbs) seed (lift_equivocators_sub_label_to full_replay_state l)  (lift_equivocators_sub_state_to full_replay_state s, om).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
∀ (l : label) (s : state) (om : option message),
  input_valid SeededXE l (s, om)
  → no_equivocations_additional_constraint_with_pre_loaded
      equivocator_IM (free_constraint equivocator_IM)
      (equivocator_Hbs IM Hbs) seed
      (lift_equivocators_sub_label_to
         full_replay_state l)
      (lift_equivocators_sub_state_to
         full_replay_state s, om)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
∀ (l : label) (s : state) (om : option message),
  input_valid SeededXE l (s, om)
  → no_equivocations_additional_constraint_with_pre_loaded
      equivocator_IM (free_constraint equivocator_IM)
      (equivocator_Hbs IM Hbs) seed
      (lift_equivocators_sub_label_to
         full_replay_state l)
      (lift_equivocators_sub_state_to
         full_replay_state s, om)
  split; [|exact I].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
l: label
s: state
om: option message
H: input_valid SeededXE l (s, om)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  destruct om as [m|]; [|exact I].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
l: label
s: state
m: message
H: input_valid SeededXE l (s, Some m)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  destruct H as [Hs [_ [_ [Hc1 _]]]].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
l: label
s: state
m: message
Hs: valid_state_prop SeededXE s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  apply (VLSM_incl_valid_state (NoEquivocation.seeded_no_equivocation_incl_preloaded equivocator_IM (free_constraint _) (equivocator_Hbs IM Hbs) seed)) in Hfull_replay_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
Hs: valid_state_prop SeededXE s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  specialize (valid_state_project_preloaded_to_preloaded _ equivocator_IM (free_constraint _) full_replay_state)
    as Hfull_replay_state_pr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
Hs: valid_state_prop SeededXE s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  pose (no_equivocations_additional_constraint_with_pre_loaded
          sub_equivocator_IM (free_constraint sub_equivocator_IM)  (equivocator_Hbs sub_IM Hbs_sub) seed)
        as constraint.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
Hs: valid_state_prop SeededXE s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  specialize
    (pre_loaded_vlsm_incl_pre_loaded_with_all_messages
      (composite_vlsm sub_equivocator_IM constraint)
      seed) as Hincl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
Hs: valid_state_prop SeededXE s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  apply (VLSM_incl_valid_state Hincl) in Hs.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  specialize (valid_state_project_preloaded_to_preloaded _ sub_equivocator_IM constraint s)
    as Hs_pr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  simpl in Hc1.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hc1: composite_no_equivocations_except_from
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed l 
  (s, Some m)
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  destruct Hc1 as [Hsub_sent | Hseeded].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hsub_sent: composite_has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM)
     equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM
        equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) 
  (s, Some m).1 m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hseeded: seed m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hsub_sent: composite_has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM)
     equivocating)
  (equivocator_Hbs
     (SubProjectionTraces.sub_IM IM
        equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) 
  (s, Some m).1 m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m) destruct Hsub_sent as [sub_i Hsent].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
sub_i: sub_index equivocating
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     sub_i) 
  ((s, Some m).1 sub_i) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
    destruct_dec_sig sub_i i Hi Heqsub_i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
sub_i: sub_index equivocating
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     sub_i) 
  ((s, Some m).1 sub_i) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m)
    left.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
sub_i: sub_index equivocating
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     sub_i) 
  ((s, Some m).1 sub_i) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
composite_has_been_sent equivocator_IM
  (equivocator_Hbs IM Hbs)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m).1 m exists i.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
sub_i: sub_index equivocating
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     sub_i) 
  ((s, Some m).1 sub_i) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
has_been_sent (equivocator_IM i)
  ((lift_equivocators_sub_state_to full_replay_state s,
   Some m).1 i) m
    simpl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
sub_i: sub_index equivocating
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     sub_i) 
  ((s, Some m).1 sub_i) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
equivocator_has_been_sent (IM i)
  (lift_equivocators_sub_state_to full_replay_state s
     i) m rewrite (lift_equivocators_sub_state_to_sub _ _ _ Hi).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
sub_i: sub_index equivocating
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     sub_i) 
  ((s, Some m).1 sub_i) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
i: index
Hi: sub_index_prop equivocating i
Heqsub_i: sub_i =
dec_exist 
  (sub_index_prop equivocating) i Hi
equivocator_has_been_sent (IM i)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)))
  m
    subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
i: index
Hi: sub_index_prop equivocating i
Hsent: has_been_sent
  (SubProjectionTraces.sub_IM
     (Common.equivocator_IM IM) equivocating
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
  ((s, Some m).1
     (dec_exist 
        (sub_index_prop equivocating) i Hi)) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
equivocator_has_been_sent (IM i)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)))
  m unfold SubProjectionTraces.sub_IM in Hsent.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
i: index
Hi: sub_index_prop equivocating i
Hsent: has_been_sent
  (Common.equivocator_IM IM
     (`
      (dec_exist 
         (sub_index_prop equivocating) i Hi)))
  ((s, Some m).1
     (dec_exist 
        (sub_index_prop equivocating) i Hi)) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
equivocator_has_been_sent (IM i)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)))
  m cbn in Hsent |-*.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
i: index
Hi: sub_index_prop equivocating i
Hsent: equivocator_has_been_sent 
  (IM i)
  (s
     (dec_exist 
        (sub_index_prop equivocating) i Hi)) m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
equivocator_has_been_sent (IM i)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)))
  m
    specialize (Hfull_replay_state_pr i Hfull_replay_state).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
i: index
Hi: sub_index_prop equivocating i
Hsent: equivocator_has_been_sent 
  (IM i)
  (s
     (dec_exist 
        (sub_index_prop equivocating) i Hi)) m
Hfull_replay_state_pr: valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (equivocator_IM i))
  (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
equivocator_has_been_sent (IM i)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)))
  m
    specialize (Hs_pr (dec_exist (sub_index_prop equivocating) i Hi) Hs).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
i: index
Hi: sub_index_prop equivocating i
Hsent: equivocator_has_been_sent 
  (IM i)
  (s
     (dec_exist 
        (sub_index_prop equivocating) i Hi)) m
Hfull_replay_state_pr: valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (equivocator_IM i))
  (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (sub_equivocator_IM
        (dec_exist
         (sub_index_prop equivocating) i Hi)))
  (s
     (dec_exist 
        (sub_index_prop equivocating) i Hi))
equivocator_has_been_sent (IM i)
  (equivocator_state_append (full_replay_state i)
     (s (dec_exist (sub_index_prop equivocating) i Hi)))
  m
    apply equivocator_state_append_sent_right; assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hseeded: seed m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
composite_no_equivocations_except_from equivocator_IM
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  Some m) right.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (free_composite_vlsm
         equivocator_IM))))
  full_replay_state
l: label
s: state
m: message
constraint:= no_equivocations_additional_constraint_with_pre_loaded
  sub_equivocator_IM
  (free_constraint sub_equivocator_IM)
  (equivocator_Hbs sub_IM Hbs_sub) seed: composite_label sub_equivocator_IM
→ composite_state sub_equivocator_IM *
  option message → Prop
Hs: valid_state_prop
  (mk_vlsm
     (machine
        (pre_loaded_with_all_messages_vlsm
         (composite_vlsm sub_equivocator_IM
         constraint)))) s
Hseeded: seed m
Hfull_replay_state_pr: ∀ i : index,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm
         equivocator_IM
         (free_constraint
         equivocator_IM)))
    full_replay_state
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (equivocator_IM i))
      (full_replay_state i)
Hincl: VLSM_incl
  (pre_loaded_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint) seed)
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm sub_equivocator_IM
        constraint))
Hs_pr: ∀ i : sub_index equivocating,
  valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (composite_vlsm sub_equivocator_IM
         constraint)) s
  → valid_state_prop
      (pre_loaded_with_all_messages_vlsm
         (sub_equivocator_IM i)) 
      (s i)
seed m assumption.
Qed.

Local Lemma sent_are_valid
  : forall m, seed m -> valid_message_prop SeededAllXE m.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
∀ m : message,
  seed m → valid_message_prop SeededAllXE m
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
∀ m : message,
  seed m → valid_message_prop SeededAllXE m
  intros m Hm.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
m: message
Hm: seed m
valid_message_prop SeededAllXE m
  apply initial_message_is_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
m: message
Hm: seed m
initial_message_prop m
  right.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
m: message
Hm: seed m
seed m assumption.
Qed.

Here we specialize the generic lift_equivocators_sub_weak_projection result for the equivocators_no_equivocations_constraint.

Lemma SeededXE_SeededNoEquiv_weak_full_projection
  : VLSM_weak_full_projection SeededXE SeededAllXE (lift_equivocators_sub_label_to full_replay_state) (lift_equivocators_sub_state_to full_replay_state).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
VLSM_weak_full_projection SeededXE SeededAllXE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
VLSM_weak_full_projection SeededXE SeededAllXE
  (lift_equivocators_sub_label_to full_replay_state)
  (lift_equivocators_sub_state_to full_replay_state)
  constructor.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
∀ (sX : state) (trX : list transition_item),
  finite_valid_trace_from SeededXE sX trX
  → finite_valid_trace_from SeededAllXE
      (lift_equivocators_sub_state_to
         full_replay_state sX)
      (pre_VLSM_full_projection_finite_trace_project
         (type SeededXE) (type SeededAllXE)
         (lift_equivocators_sub_label_to
            full_replay_state)
         (lift_equivocators_sub_state_to
            full_replay_state) trX)
  apply lift_equivocators_sub_weak_projection; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
i: index
ns: vstate (IM i)
s: state
H: i ∈ equivocating
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed) s
no_equivocations_additional_constraint_with_pre_loaded
  equivocator_IM (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed (existT i (Spawn ns))
  (s, None)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
m: message
H: seed m
valid_message_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed) m
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
  full_replay_state
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors
  IM
→ ∀ i : index,
    MachineDescriptor
      (IM i)
    → ∀ j : index,
        MachineDescriptor
         (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
         IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
         IM))
  → option
      (list
         (composite_transition_item
         IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
l: label
s: state
om: option message
H: input_valid SeededXE l (s, om)
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
  (lift_equivocators_sub_state_to
     full_replay_state s)
no_equivocations_additional_constraint_with_pre_loaded
  equivocator_IM 
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
i: index
ns: vstate (IM i)
s: state
H: i ∈ equivocating
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed) s
no_equivocations_additional_constraint_with_pre_loaded
  equivocator_IM (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed (existT i (Spawn ns))
  (s, None) split; exact I.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
m: message
H: seed m
valid_message_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
           equivocator_IM
           (free_constraint equivocator_IM)
           (equivocator_Hbs IM Hbs) seed)) seed) m apply sent_are_valid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
m: message
H: seed m
seed m assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
           equivocator_IM
           (free_constraint equivocator_IM)
           (equivocator_Hbs IM Hbs) seed)) seed)
  full_replay_state assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
l: label
s: state
om: option message
H: input_valid SeededXE l (s, om)
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
  (lift_equivocators_sub_state_to
     full_replay_state s)
no_equivocations_additional_constraint_with_pre_loaded
  equivocator_IM (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om) apply SeededNoEquiv_subsumption; assumption.
Qed.

Lemma sub_replayed_trace_from_valid_equivocating
  (is : composite_state sub_equivocator_IM)
  (tr : list (composite_transition_item sub_equivocator_IM))
  (Htr : finite_valid_trace SeededXE is tr)
  : finite_valid_trace_from SeededAllXE
      full_replay_state (replayed_trace_from full_replay_state is tr).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= composite_no_equivocation_vlsm_with_pre_loaded
  equivocator_IM
  (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  unfold composite_no_equivocation_vlsm_with_pre_loaded in SeededAllXE.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  specialize
    (sub_preloaded_replayed_trace_from_valid_equivocating
      (no_equivocations_additional_constraint_with_pre_loaded equivocator_IM (free_constraint _) (equivocator_Hbs IM Hbs) seed)
      seed)
      as Hvalid.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: (∀ (i : index) 
   (ns : vstate (IM i)) 
   (s : state),
   i ∈ equivocating
   → valid_state_prop
       (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
       s
     → no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (existT i (Spawn ns)) 
         (s, None))
→ (∀ m : message,
     seed m
     → valid_message_prop
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         m)
  → ∀ full_replay_state : 
       composite_state equivocator_IM,
      valid_state_prop
        (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
        full_replay_state
      → (∀ (l : label) 
         (s : state) 
         (om : option message),
         input_valid SeededXE l (s, om)
         → 
         valid_state_prop
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         (lift_equivocators_sub_state_to
         full_replay_state s)
         → 
         no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
        → 
        ∀ (is : 
         composite_state sub_equivocator_IM) 
         (tr : 
         list
         (composite_transition_item
         sub_equivocator_IM)),
         finite_valid_trace SeededXE is tr
         → 
         finite_valid_trace_from
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         full_replay_state
         (replayed_trace_from
         full_replay_state is tr)
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  spec Hvalid; [split; exact I|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: (∀ m : message,
   seed m
   → valid_message_prop
       (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
       m)
→ ∀ full_replay_state : 
     composite_state equivocator_IM,
    valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
      full_replay_state
    → (∀ (l : label) 
         (s : state) 
         (om : option message),
         input_valid SeededXE l (s, om)
         → 
         valid_state_prop
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         (lift_equivocators_sub_state_to
         full_replay_state s)
         → 
         no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
      → ∀ (is : 
         composite_state sub_equivocator_IM) 
         (tr : 
         list
         (composite_transition_item
         sub_equivocator_IM)),
         finite_valid_trace SeededXE is tr
         → 
         finite_valid_trace_from
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         full_replay_state
         (replayed_trace_from
         full_replay_state is tr)
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  spec Hvalid; [apply sent_are_valid|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: ∀ full_replay_state : 
   composite_state equivocator_IM,
  valid_state_prop
    (pre_loaded_vlsm
       (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
    full_replay_state
  → (∀ (l : label) 
       (s : state) 
       (om : option message),
       input_valid SeededXE l (s, om)
       → valid_state_prop
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         (lift_equivocators_sub_state_to
         full_replay_state s)
         → 
         no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
    → ∀ (is : 
         composite_state sub_equivocator_IM) 
        (tr : 
         list
         (composite_transition_item
         sub_equivocator_IM)),
        finite_valid_trace SeededXE is tr
        → 
        finite_valid_trace_from
         (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
         full_replay_state
         (replayed_trace_from
         full_replay_state is tr)
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  specialize (Hvalid _ Hfull_replay_state).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: (∀ (l : label) 
   (s : state) 
   (om : option message),
   input_valid SeededXE l (s, om)
   → valid_state_prop
       (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
       (lift_equivocators_sub_state_to
         full_replay_state s)
     → no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
→ ∀ (is : composite_state sub_equivocator_IM) 
    (tr : 
     list
       (composite_transition_item
         sub_equivocator_IM)),
    finite_valid_trace SeededXE is tr
    → finite_valid_trace_from
        (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
        full_replay_state
        (replayed_trace_from
         full_replay_state is tr)
finite_valid_trace_from SeededAllXE full_replay_state
  (replayed_trace_from full_replay_state is tr)
  apply Hvalid; [|assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: (∀ (l : label) 
   (s : state) 
   (om : option message),
   input_valid SeededXE l (s, om)
   → valid_state_prop
       (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
       (lift_equivocators_sub_state_to
         full_replay_state s)
     → no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
→ ∀ (is : composite_state sub_equivocator_IM) 
    (tr : 
     list
       (composite_transition_item
         sub_equivocator_IM)),
    finite_valid_trace SeededXE is tr
    → finite_valid_trace_from
        (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
        full_replay_state
        (replayed_trace_from
         full_replay_state is tr)
∀ (l : label) (s : state) (om : option message),
  input_valid SeededXE l (s, om)
  → valid_state_prop
      (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
            (no_equivocations_additional_constraint_with_pre_loaded
               equivocator_IM
               (free_constraint equivocator_IM)
               (equivocator_Hbs IM Hbs) seed)) seed)
      (lift_equivocators_sub_state_to
         full_replay_state s)
    → no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed
        (lift_equivocators_sub_label_to
           full_replay_state l)
        (lift_equivocators_sub_state_to
           full_replay_state s, om)
  intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: (∀ (l : label) 
   (s : state) 
   (om : option message),
   input_valid SeededXE l (s, om)
   → valid_state_prop
       (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
       (lift_equivocators_sub_state_to
         full_replay_state s)
     → no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
→ ∀ (is : composite_state sub_equivocator_IM) 
    (tr : 
     list
       (composite_transition_item
         sub_equivocator_IM)),
    finite_valid_trace SeededXE is tr
    → finite_valid_trace_from
        (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
        full_replay_state
        (replayed_trace_from
         full_replay_state is tr)
l: label
s: state
om: option message
H: input_valid SeededXE l (s, om)
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
  (lift_equivocators_sub_state_to
     full_replay_state s)
no_equivocations_additional_constraint_with_pre_loaded
  equivocator_IM (free_constraint equivocator_IM)
  (equivocator_Hbs IM Hbs) seed
  (lift_equivocators_sub_label_to full_replay_state l)
  (lift_equivocators_sub_state_to full_replay_state s,
  om)
  apply SeededNoEquiv_subsumption.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
seed: message → Prop
equivocating: list index
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equiv_index:= sub_index equivocating: Type
equivocator_descriptors:= Common.equivocator_descriptors
  IM: Type
equivocators_state_project:= Common.equivocators_state_project
  IM: Common.equivocator_descriptors
  IM
→ vstate
    (free_composite_vlsm
       (Common.equivocator_IM
          IM))
  → vstate
      (free_composite_vlsm
         IM)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
sub_equivocator_IM:= SubProjectionTraces.sub_IM
  equivocator_IM equivocating: sub_index equivocating
→ VLSM message
sub_IM:= SubProjectionTraces.sub_IM IM equivocating: sub_index equivocating → VLSM message
equivocator_descriptors_update:= Common.equivocator_descriptors_update
  IM: Common.equivocator_descriptors IM
→ ∀ i : index,
MachineDescriptor (IM i) → ∀ j : index, MachineDescriptor (IM j)
proper_equivocator_descriptors:= Common.proper_equivocator_descriptors
  IM: Common.equivocator_descriptors IM
→ vstate (free_composite_vlsm (Common.equivocator_IM IM)) → Prop
equivocators_trace_project:= Projections.equivocators_trace_project
  IM: Common.equivocator_descriptors
  IM
→ list
    (composite_transition_item
       (Common.equivocator_IM
          IM))
  → option
      (list
         (composite_transition_item
            IM) *
       Common.equivocator_descriptors
         IM)
Free:= free_composite_vlsm IM: VLSM message
FreeE:= free_composite_vlsm equivocator_IM: VLSM message
PreFreeE:= pre_loaded_with_all_messages_vlsm FreeE: VLSM message
FreeSubE:= free_composite_vlsm sub_equivocator_IM: VLSM message
PreFreeSubE:= pre_loaded_with_all_messages_vlsm
  FreeSubE: VLSM message
Hbs_sub:= sub_has_been_sent_capabilities IM
  equivocating Hbs: ∀ sub_i : sub_index equivocating,
  HasBeenSentCapability (sub_IM sub_i)
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  Hbs equivocating seed: VLSM message
equivocators_no_equivocations_vlsm:= Common.equivocators_no_equivocations_vlsm
  IM Hbs: VLSM message
SeededAllXE:= pre_loaded_vlsm
  (composite_vlsm equivocator_IM
     (no_equivocations_additional_constraint_with_pre_loaded
        equivocator_IM
        (free_constraint equivocator_IM)
        (equivocator_Hbs IM Hbs) seed))
  seed: VLSM message
full_replay_state: composite_state equivocator_IM
Hfull_replay_state: valid_state_prop SeededAllXE
  full_replay_state
is: composite_state sub_equivocator_IM
tr: list
  (composite_transition_item sub_equivocator_IM)
Htr: finite_valid_trace SeededXE is tr
Hvalid: (∀ (l : label) 
   (s : state) 
   (om : option message),
   input_valid SeededXE l (s, om)
   → valid_state_prop
       (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
       (lift_equivocators_sub_state_to
         full_replay_state s)
     → no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed
         (lift_equivocators_sub_label_to
         full_replay_state l)
         (
         lift_equivocators_sub_state_to
         full_replay_state s, om))
→ ∀ (is : composite_state sub_equivocator_IM) 
    (tr : 
     list
       (composite_transition_item
         sub_equivocator_IM)),
    finite_valid_trace SeededXE is tr
    → finite_valid_trace_from
        (pre_loaded_vlsm
         (composite_vlsm equivocator_IM
         (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
        full_replay_state
        (replayed_trace_from
         full_replay_state is tr)
l: label
s: state
om: option message
H: input_valid SeededXE l (s, om)
H0: valid_state_prop
  (pre_loaded_vlsm
     (composite_vlsm equivocator_IM
        (no_equivocations_additional_constraint_with_pre_loaded
         equivocator_IM
         (free_constraint equivocator_IM)
         (equivocator_Hbs IM Hbs) seed)) seed)
  (lift_equivocators_sub_state_to
     full_replay_state s)
input_valid SeededXE l (s, om)
  assumption.
Qed.

End seeded_no_equiv.

End all_equivocating.